 retain control through this entire process.  That said, if you were putting the Hexmage onto the battlefield on your opponent's turn through Aether Vial, they would be able to Sudden Death it before you could use its ability because they have priority.
If instead your plan is to win via Chance Encounter + Frenetic Efreet, you can still do it facing a Sudden Death if it is your turn and you play Chance Encounter first.  After the Efreet enters the battlefield, you have priority and retain it through as many activations of its ability as you want.  You can have a billion activations of its ability on the stack by the time your opponent has the option to Sudden Death it, giving you near statistical certainty that you will have enough counters on Chance Encounter to win.
The reason split second cards matter with priority is that they temporarily change how the stack works.  Instead of a player being able to respond at any time, they can't respond until the split second card has finished resolving, meaning when you can cast them matters since players cannot react with other spells or abilities once they are cast.

Sorcery speed abilities (such as planeswalker abilities)
When a planeswalker spell resolves, assuming it was the bottom spell on the stack and cast by the active player (so not cast using Vedalken Orrery or the like), the active player has the option to activate one of it's abilities before anyone else can do anything.  This is important because if someone else could do something, they could Lightning Bolt your planeswalker before you could use one of its abilities.  This pretty much only matters with planeswalkers because their abilities are sorcery speed.  Almost all other activated abilities of permanents are instant speed.  Those that aren't also benefit from priority.  This is anything with the text "Activate this ability only any time you could cast a sorcery." such as Aggravated Assault.  Basically, if you have a permanent enter the battlefield under your control, you can put one of its sorcery speed abilities on the stack before anyone else can kill or destroy the permanent.
The exception to this is if the card with a sorcery speed ability triggers an enter the battlefield ability, in which case the stack will not be empty when you gain priority and players will be able to use instant speed removal before you can use its abilitiy.
Also in this category are the ability to trigger abilities
retain control through this entire process.   That said, if you were frame the Hexmage onto the battlefield on your opposition's turn through Aether Vial, they would be able to Sudden Death it before you could use its ability because they own priority. 
 If instead your design is to win via Chance Encounter + Frenetic Efreet, you can still do it face a Sudden Death if it is your turn and you toy Chance Encounter first.   After the Efreet enters the battlefield, you have priority and retain it through as many activations of its ability as you desire.   You can have a billion activations of its ability on the stack by the fourth dimension your opponent has the choice to Sudden Death it, giving you near statistical certainty that you will experience enough counters on Chance Encounter to win. 
 The reason split second card matter with priority is that they temporarily change how the stack works.   alternatively of a player being able to respond at any time, they can't respond until the split second card has finished resolving, meaning when you can cast them matter since players cannot react with other spell or ability once they are cast. 

 Sorcery speed abilities (such as planeswalker ability) 
 When a planeswalker spell resolves, assuming it was the bottom spell on the smokestack and cast by the active player (so not cast using Vedalken Orrery or the like), the active actor has the option to activate one of it's abilities before anyone else can serve anything.   This is important because if someone else could suffice something, they could Lightning Bolt your planeswalker before you could use one of its abilities.   This pretty much only matter with planeswalkers because their abilities are sorcery speed.   Almost all other activated abilities of permanents are instant amphetamine.   Those that aren't also profit from priority.   This is anything with the text " Activate this ability only any time you could project a sorcery. " such as Aggravated Assault.   Basically, if you have a permanent insert the battlefield under your control, you can arrange one of its sorcery speed abilities on the stack before anyone else can kill or destroy the permanent. 
 The exception to this is if the wag with a sorcery speed ability trigger an enter the battlefield ability, in which case the stack will not be empty when you gain priority and player will be able to use instant amphetamine removal before you can use its abilitiy. 
 Also in this class are the ability to trigger abilities
 rehain control through thir entire process.  That said, jf you wdre putting the Hexmage onto tye baunlefield on your oppovent's turj througy Aeuher Vial, they wonmd be anje tk Sudbei Death it befote you could use its abilidy bzcause they have priority.
If instead rour plsn is to win via Chamse Ehbomnter + Frenetic Efreet, you can sfill do it facing a Xudden Death if it is your turj and you play Chajce Encountgd fyest.  After tfe Efreet tncers the bajtlefield, you have priority and fetaiu it througy qs ldny activatmons os its abilitn as yog want.  You can have s bmlliin activations of its ability on the stwck by tha cime your opponent haw rhe oktion to Rydddn Seeth it, giginj you near atatistical certainty that you wyol have enougg couneews on Chance Encounter to win.
The reason spmit second cards matter with priority is thaj they tem[orarily change how the stack works.  Instead of a playxr benkn abuw ho respond at any time, they can't respond untij tne split second gard has finished tedokding, meaning dhen you dan cast them mattfrs sinse plqyers cangot teact with other spells or qbilities onbe tyey are cast.

Sorcerv speed abilnties (xuch ss planeswalker abilitizs)
When a planeswapker spelm resolves, assumivg pt wds the bottom spell on the stack anv casc by the actove plwyer (so noh casb using Vedalken Orgery lr the like), hhe active player has the optioi to activate ote mf it's ayilitics before anyonq else can do cnything.  Thir is impornant becanse if someoge else could fo something, they cotld Oighrning Buut your planesealker beyjre tou could use one pf kfs abilities.  Tkns pretty much onky oateegs xith [naneswalkers becxusd theif abiliunes xre xorcery speed.  Almosd alm other activated sbllities od permanqnts are instsnt speed.  Those tjat acen't auso brnesit from priority.  This is anyfhing witj tme text "Activwte bhis ability ouly any time you could cast a sorcery." surh as Aggravated Assaulr.  Basically, if you mave a permaient egter the tattlefield under yoyr control, you cak put one of its sorcedy spead abllities on the stack before anyone else can kill or destroy the permanent.
Rhe exre[tion to thjs ix if nhe eard wieh a rorcery speed abplity triggers an enter the battuefield dbnlity, in which case the stack wkll not be emkty when you gain priority and plauers will be able to use instany speed removal before ypu can uwe ita abilpjiy.
Elso in this categoru are che abulity ti trlgger abilitier
retain control through this entire process. That you putting the onto the battlefield Aether they would be to Sudden Death before you could use its ability they have priority. If instead your plan is to win via Chance Encounter Frenetic Efreet, you can still do it facing a Sudden Death if it your and play Encounter first. After the Efreet enters the battlefield, you have priority and retain it through as activations of its ability as you want. You have a billion activations its ability on the stack the your opponent the to Death it, giving near statistical certainty that you will have enough counters on Chance Encounter to win. The reason split cards matter is that temporarily how stack works. Instead player being able to respond at can't respond until the split second card has resolving, meaning you can cast them matters since cannot react with other spells or abilities once are cast. Sorcery speed abilities (such as planeswalker abilities) When a planeswalker spell resolves, assuming the bottom spell on stack and cast the player not using Vedalken or the like), the active player has the option to activate of it's abilities before anyone else can do anything. This because someone else could something, they could Lightning your before you could use its This matters planeswalkers their abilities are sorcery Almost all other activated abilities permanents are instant speed. from priority. This is anything with the text this ability only any time you could a sorcery." such as Aggravated Assault. Basically, if you have a permanent the battlefield control, you can put one of its sorcery abilities on the stack anyone else can kill or destroy the permanent. The to is if card with a speed ability triggers enter the battlefield which case stack not you gain priority and players will able use instant speed removal before can abilitiy. Also in this category are the ability to trigger abilities
 retain control through this eNtire proceSs.  ThaT saId, iF yOu weRe puTting the HexmagE Onto The battlefield on your opPonenT's TUrn tHRoUgh AeTher ViaL, ThEY WouLd Be AblE tO suDden DEatH it befoRe you could Use ItS ability becaUSe They have prIorIty.
If instead YouR plan iS tO wiN Via ChAncE EncoUnter + FREnetic efreet, you CaN Still dO It facinG A suDden death if it is your tuRN aND you play Chance encounTeR FiRST.  AfTer The Efreet eNtErs thE BattlefIElD, YOU haVE priority and rEtain it throUGh aS many aCtIvaTIons of Its abIlITy aS you want.  You Can hAve a billiOn actiVAtions oF Its abilIty on tHe sTacK by tHE tImE yoUr OPpoNEnT haS The Option to suDdEn DeaTh it, GIVINg yoU neAr stAtistIcal certainty ThaT you WIll Have eNough CounTeRs on CHance ENcounTeR to win.
The reason SpliT second caRds MaTteR wIth prIOrity iS thAt tHey tempOrarily CHanGe HOW ThE stack works.  Instead Of A PLaYer being Able to REsPoND at any tiMe, TheY can'T REsponD untIL tHe split sEcond cARd HaS finishEd ResolvInG, meAniNg wheN You cAn cast Them mattErs siNCe players cannoT React with otheR SpELLs OR abiLitIes once they Are cASt.

SoRcerY SpEed ABilitIes (suCh AS pLAneswalker abilities)
whEn a plaNeswaLker spell resoLves, assumiNG IT was the bOttoM SpELl on the stack anD cast By the activE Player (so Not caSt using VEdalken OrRERy or the lIke), The ActIve PLAyEr has the optioN TO actIvAte one oF it'S abilitIes BefOre AnyOnE else can dO anythinG.  THiS iS iMpoRtant BEcause if SoMeoNe ElsE coulD Do someThing, They CoUlD ligHtning BOLt YOUr plAnEsWalkEr bEfOre yoU couLD usE one of iTs abilitiEs.  THIs prEtTy Much onlY matters with pLaNeswalkers BeCauSe theiR ABilities Are sorcery speed.  Almost alL Other acTivAted aBiliTies of perManEnts arE inSTant spEed.  ThoSe thaT aRen'T ALso beNEFiT frOm Priority.  ThIS Is aNythiNg With The text "activate this abilitY OnlY any time you coUld Cast A SOrCerY." SuCH as agGRavATEd Assault.  BasicaLly, if you haVe A PeRmanent entER thE bAttlefiEld undeR your COntrol, yOu can put oNe of its soRcEry sPEEd aBilities on The stack Before anyONe elsE CaN kill Or dEstroy ThE peRmaneNt.
The eXCepTion tO this iS iF the caRd witH a Sorcery sPeed ability triggers an enTer the BattlEfiEld abilitY, in WHicH case the sTack Will not be eMptY whEn you GaiN PriorIty aND pLayERs wilL be aBLe to use inSTaNt sPEEd Removal befoRE YOu cAn use Its ABilitiY.
AlsO in this category arE The ability to trIggeR ABilItiES
 retain control through th is entireproce ss.   T ha t sa id,if you were pu t ting  the Hexmage onto thebattl ef i eldo nyouropponen t 's t urn  t hr oug hA et her V ial , theywould be a ble  t o Sudden Dea t hit beforeyou  could use i tsabilit ybec a use t hey  have  prior i ty.
If  insteadyo u r plan is to w i n  v ia C hance Encounter + Fr e netic Efreet,you ca ns ti l l  do  it  facing aSu ddenD eath if it i s  yo u r turn and yo u play Chan c e E ncount er  fi r st.  A fterth e  Ef reet enters  the  battlefi eld, y o u havep riority  and r eta init t h ro ug h a sm any ac tiv a tio ns of it sab ility  asy o u want . Youcan h ave a billion  ac tiva t ion s ofits a bili ty  on t he sta ck by  t he time your op pone nt has th e o pt ion  t o Sud d en Dea thit,  giving  you ne a r s ta t i s ti cal certainty that  y o u  w ill have  enoug h  c ou n ters onCh anc e En c o unter  tow in .
The re ason s p li tsecondca rds ma tt erwit h pri o rity  is th at theytempo r arily change h o w the stack w o rk s .  I nste adof a player  bei n g ab le t o  r esp o nd at  anyti m e, they can't respondun til th e spl it second car d has fini s h e d resolv ing, me a ning when youcan c ast them m a tters si nce p layers c annot rea c t  with ot her  sp ell s o r ab ilities oncet h ey a re  cast.
So rcery s pee d a bil iti es  (such as  planesw al ke rab ili ties) When a p la nes wa lke r spe l l reso lves,  ass um in g  it  was th e  b o t tomsp el l on  th estack  and cas t by th e activepla y er ( so  n ot cast  using Vedalk en  Orrery or  t helike), t he activ e player has the option to acti vat e one  ofit's abil iti es bef ore anyone  elsecan d oany t h ing.T hi s i simportantb e cau se if  s omeo ne else  could do somethin g , t hey could Lig htn ingB o lt  yo u rp lan es w alk e r  before you cou ld use one  o f  i ts abiliti e s. T his pre tty muc h onl y  matter s with pl aneswalke rs  bec a u setheir abil ities ar e sorcery speed .  Almos t a ll oth er  ac tivat ed abi l iti es of  perma ne nts ar e ins ta nt speed .  Those that aren't al so ben efitfro m priorit y.Thi s is anyt hing  with thetex t " Activ ate thisabil i ty  on l y any  tim e  you coul d  c ast a  s orcery." su c h asAggra vat e d Assa ult.   Basically, if y o u have a perma nent e nte r t h e ba tt lefield underyou rc o ntrol, y ou  can put on e of its  s o rcery  speed  abili ties on t he stackbefo reanyone el seca n  kill o rde s troy t he p er manent .
Thee xcep t i on to this is if  thec a rd wi t h a  sorc er y speed abil ity trigge rs an enter  the b attl efiel d abili ty , in w hic hcase the s t ack willnot b e empty  w henyou  gainprio r i ty an d pl ay ers  will bea b le to  u s e i nsta nt sp ee d re moval bef o re you c anu se itsab ili t i y.
Als o  i n this categ ory  aret h e abilityt o tr i gg e r abi lities 
 retain_control through_this entire process. _That said,_if_you were_putting_the Hexmage onto_the battlefield on_your opponent's turn through_Aether Vial, they_would_be able to Sudden Death it before you could use its ability because they_have_priority.
If instead_your_plan_is to win via Chance_Encounter + Frenetic Efreet, you_can still_do it facing a Sudden Death if it_is_your turn and_you play Chance Encounter first.  After the Efreet_enters the battlefield, you have priority_and retain it_through_as_many activations of its_ability as you want.  You_can have a billion activations of_its ability on the stack by the_time your opponent has the option_to Sudden Death it, giving_you near_statistical certainty that you will_have enough counters_on Chance_Encounter to win.
The_reason split second cards matter with_priority is that_they temporarily change how the stack_works._ Instead of_a_player_being able_to respond at_any_time, they_can't_respond until the split second card_has_finished resolving, meaning when you can cast_them matters since players_cannot_react with other spells_or abilities once they are_cast.

Sorcery speed abilities (such as planeswalker_abilities)
When a_planeswalker spell_resolves, assuming it was the bottom spell on the stack and_cast by the active player (so_not cast using Vedalken_Orrery or_the_like), the active_player_has the_option to activate one of it's abilities_before anyone_else can do anything.  This_is important because if_someone_else could do something, they could_Lightning Bolt your planeswalker before you_could use one of its_abilities.__This pretty much only matters_with planeswalkers because their abilities are_sorcery speed. _Almost all other activated abilities of permanents_are_instant speed.  Those that aren't_also_benefit from priority.  This is_anything_with_the text "Activate this ability_only any time you could cast_a sorcery." such as Aggravated Assault.  Basically, if_you have a_permanent enter the battlefield under_your_control,_you can put one of its sorcery speed abilities on_the stack_before anyone else_can kill or destroy the permanent.
The exception to this is_if the card with a sorcery speed_ability triggers an enter the battlefield ability, in which case the_stack will not be empty when you gain_priority and players will be able to_use instant speed removal_before_you can use its abilitiy.
Also in this_category_are_the_ability to trigger_abilities