 the fact that 'Tom is in Paris'. Notice that there is a non-finite subordinate clause here.

Maria is used to [ Tom being in Paris ].

In the subordinate clause, the verb is being. This is the  --ing form of the verb BE. It is non-finite because it does not have any tense. It is not a present or past form of the verb. The Subject of this verb is Tom. Now look at the sentence below:

Maria is used to [ being in Paris ].

Here the subordinate clause also has the --ing for of the verb BE.  But there is no subject for this verb. However, we understand that the subject of the verb BE is Maria. This is because, if the Subject of the verb used and the Subject of the subordinate clause is the same person or thing, we do not need to repeat it. We can understand the sentence like this:

Mariai is used to [ ____i being in Paris ]

We can see that there is a gap in this sentence, where we expect the subject of the phrase "being in Paris" to be. The identity of this gap is controlled by the subject of the verb used.
In other words we interpret the sentence like this:
Maria is used to [ Maria being in Paris ].
This just means that Maria is used to the fact that Maria is in Paris.
The Original Poster's Question
Let's look at example 1:

She is used to there being no one else around.

Here the phrase be around just means to be close, or to be nearby. So if I live with my brother, I can say that "my brother is often around". This just means that he is often in the house with me. He is often doing things in the house. In sentence (1), we can see the same structure as in the examples above. We have the verb used and a subordinate clause:

She is used to  [there being no one else around ].

This means that she is used to the fact that There is nobody else around. In the example we see the --ing form of the verb BE. The subject of this verb is the dummy pronoun there. It is the same word we see in existential sentences like there is a new new president. The sentence is perfectly grammatical. "Nobody else around" means that she is the only person around. There is nobody apart from her in that
the fact that' Tom is in Paris'. Notice that there is a non - finite subordinate clause here. 

 Maria is use to [ Tom being in Paris ]. 

 In the dependent clause, the verb is being. This is the   --ing form of the verb BE. It is non - finite because it does not get any tense. It is not a present or past shape of the verb. The Subject of this verb is Tom. Now expect at the prison term below: 

 Maria is use to [ being in Paris ]. 

 Here the subordinate clause besides has the --ing for of the verb BE.   But there is no subject for this verb. However, we understand that the national of the verb BE is Maria. This is because, if the Subject of the verb used and the Subject of the subordinate clause is the like person or thing, we do not need to repeat it. We can sympathize the sentence like this: 

 Mariai is used to [ _ _ _ _ i being in Paris ] 

 We can see that there is a gap in this conviction, where we expect the subject of the phrase " being in Paris " to be. The identity of this gap is controlled by the national of the verb used. 
 In other words we interpret the sentence like this: 
 Maria is used to [ Maria being in Paris ]. 
 This just means that Maria is used to the fact that Maria is in Paris. 
 The Original Poster's Question 
 Let's look at example 1: 

 She is used to there being no one else around. 

 Here the phrase be around just mean to be close, or to be nearby. So if I exist with my buddy, I can say that " my brother is often around ". This just means that he is often in the house with me. He is frequently doing things in the house. In sentence (1), we can see the same social organization as in the examples above. We have the verb used and a subordinate clause: 

 She is used to   [ there being no one else around ]. 

 This mean that she is used to the fact that There is nobody else around. In the example we see the --ing mannequin of the verb BE. The subject of this verb is the dummy pronoun there. It is the same word we examine in existential sentences like there is a new modern president. The prison term is perfectly grammatical. " Nobody else around " means that she is the lone person around. There embody nobody apart from her in that
 thf fact that 'Tom is in Pavis'. Notice that jhwre is a non-rinite sjbordinate clause here.

Maria ms uwed ti [ Tom being in Paris ].

Kn the suhordinatw cleuse, the verb is being. Tmns is bhe  --nnj form of the vgrb BE. It is non-finite becdure it does not have any tense. It is njt a prrsfnt or past fotm of ehe bvrn. The Subject of this verb is Toj. Now lmok at the semtence below:

Maria is used ho [ heing in Paris ].

Herf the suboreinaew clause alsu has the --pug for of tge verb BE.  But there is no subgect yor this vety. Hoagver, we undecstand that the sunkect ox the vrrb BE is Maris. Tiis us because, if the Suboect of the verb useq and the Sbbject of the subordibare clduse is gye ramt pxrskn or hhiig, we do nof need to rwpeat it. We can undtrseqnd the sentehce lihe this:

Mariai is used to [ ____i being in Pariv ]

Ws can see that there is a gap in this sentenfe, where re expect the subject of the phrase "being in Pariv" to ue. Tht lqdbtlty of this gap is controlled by the subject jr uhe verb used.
In obher words we intetpgey the sentence like cgia:
Maria is used to [ Maria feing in Paris ].
Thix just means that Maria is ysed to the yacr that Maria is in Paris.
The Oxiginak Posyer's Question
Let's look ct exajple 1:

She is used to ffere being no ond eksa around.

Here the phrase be around jnst mzans to ce ckose, ow to be newrby. So if I live with mj brojher, I can say tjat "my brother is often around". Viis just meanx dhan he is oyten ik the house witr me. He is oftgn doing chings in the holse. In seitence (1), we cwn see the sakg structure av in the exanplew above. De have the vetb used aub a suboreinate clause:

She ix ursd to  [there benug no one else arpuna ].

Trid kewts that she hs ured yo thd fact ukat Gherr is nobody else arognd. Jn the example we xec the --ing form of the verb BE. Yhe subject of thid veru is tie dumky kronoun there. It is the same wodd we see in existential senbencgs like thexe is a new new president. The sentence iw perfectly grammarical. "Nobody else atoukd" means thav she ys the onny person around. Theee is nobody aparu from her in that
the fact that 'Tom is in Paris'. there a non-finite clause here. Maria being Paris ]. In subordinate clause, the is being. This is the --ing of the verb BE. It is non-finite because it does not have any It is not a present or past form of the verb. The Subject this is Now at the sentence below: Maria is used to [ being in Paris ]. Here the subordinate also has the --ing for of the verb But there is no for this verb. However, we that subject of verb is This is because, the Subject of the verb used and the Subject of the subordinate clause is the same person thing, we need to it. can the sentence like is used to [ ____i being We can see that there is a gap this sentence, we expect the subject of the "being in Paris" to be. The identity of gap is controlled by the subject of the verb used. In other words we interpret like this: Maria is to [ Maria in ]. just that Maria used to the fact that Maria is in Paris. The Original Question Let's look at example 1: She is used to no else around. Here phrase be around just to close, or to be if live I say "my brother is often This just means that he often in the house doing things in the house. In sentence (1), can see the same structure as in examples above. We have the verb used and a subordinate clause: She used to no one else around ]. This means that is used to the that There is nobody else around. In the example see --ing form the verb BE. subject of this is the dummy It is same we sentences like there is a new president. sentence is perfectly grammatical. "Nobody around" she is the only person around. There is nobody apart from her in that
 the fact that 'Tom is in Paris'. NoTice that thEre is A noN-fiNiTe suBordInate clause herE.

mariA is used to [ Tom being in ParIs ].

In tHe SUborDInAte clAuse, the VErB IS beInG. THis Is THe  --Ing foRm oF the verB BE. It is non-FinItE because it doES nOt have any tEnsE. It is not a preSenT or pasT fOrm OF the vErb. the SuBject oF This veRb is Tom. NoW lOOk at thE SentencE BElOw:

MaRia is used to [ being iN paRIs ].

Here the suborDinate ClAUsE ALso Has The --ing for oF tHe verB bE.  But thERe IS NO suBJect for this veRb. However, we UNdeRstand ThAt tHE subjeCt of tHe VErb bE is Maria. ThIs is Because, if The SubJEct of thE Verb useD and thE SuBjeCt of THe SuBorDiNAte CLaUse IS thE same perSoN oR thinG, we dO NOT Need To rEpeaT it. We Can understand The SentENce Like tHis:

MaRiai Is Used tO [ ____i beinG in PaRiS ]

We can see that thEre iS a gap in thIs sEnTenCe, Where WE expecT thE suBject of The phraSE "beInG IN paRis" to be. The identity Of THIs Gap is conTrolleD By ThE Subject oF tHe vErb uSED.
In otHer wORdS we interPret thE SeNtEnce likE tHis:
MarIa Is uSed To [ MarIA beiNg in PaRis ].
This jUst meANs that Maria is uSEd to the fact thAT MARIa IS in PAriS.
The OriginaL PosTEr's QUestIOn
let'S Look aT examPlE 1:

shE Is used to there being nO oNe else ArounD.

Here the phrasE be around jUST Means to bE cloSE, oR To be nearby. So if i live With my brotHEr, I can saY that "My brotheR is often aROUnd". This jUst MeaNs tHat HE Is Often in the houSE With Me. he is oftEn dOing thiNgs In tHe hOusE. IN sentence (1), We can see ThE sAmE sTruCture AS in the exAmPleS aBovE. We haVE the veRb useD and A sUbORdiNate claUSe:

sHE is uSeD tO  [theRe bEiNg no oNe elSE arOund ].

ThiS means thaT shE Is usEd To The fact That There is noBoDy else arouNd. in tHe examPLE we see thE --ing form of the verb BE. The sUBject of ThiS verb Is thE dummy proNouN there. it iS The samE word wE see iN eXisTENtial SENtEncEs Like there iS A New New prEsIdenT. The senTence is perfectly grAMmaTical. "Nobody elSe aRounD" MEaNs tHAt SHe iS tHE onLY Person around. TheRe is nobody ApARt From her in tHAt
 the fact that 'Tom is inParis'. No ticetha t t he re i s anon-finite sub o rdin ate clause here.

Mari a isus e d to [Tom b eing in Pa r i s ] .
I n t he su bordi nat e claus e, the ver b i sbeing. Thisi sthe  --ing  fo rm of the ve rbBE. It  i s n o n-fin ite  beca use it does n ot have a ny tense. It is n o t  a  pre sent or past form of the verb. TheSubjec to ft h isver b is Tom.No w loo k  at the se n t e nce below:

Maria  is used to [ b eing i nPar i s ].
Hereth e  su bordinate c laus e also ha s the- -ing fo r  of the  verbBE.   B ut t h er eisno sub j ec t f o r t his verb .Ho wever , we u n d erst and  tha t the  subject of t heverb BEis Ma ria.This  i s bec ause,if th eSubject of theverb  used and  th eSub je ct of the su bor din ate cla use ist hesa m e pe rson or thing, wedo n ot  need to  repea t  i t. We can u nd ers tand t he se nten c elike thi s:

Ma r ia iis used  t o [ __ __ i b ein g inP aris  ]

We  can see  that there is a gap in this sente n ce , wh e re w e e xpect the s ubje c t of  the ph ras e  "bei ng in  P a ri s " to be. The identi ty  of th is ga p is controll ed by thes u b ject ofthev er b  used.
In othe r wor ds we inte r pret the  sent ence lik e this:
M a r ia is us edto[ M ari a be ing in Paris] . 
Thi sjust me ans  that M ari a i s u sed  t o the fac t that M ar ia  i sinParis . 
The Ori gi nal  P ost er'sQ uestio n
Let 's l oo ka t e xample1 :S he i sus ed t o t he re be ingn o o ne else  around.
He r e th eph rase be  around justme ans to becl ose , or t o be nearb y. So if I live with my brother , I  cansaythat "mybro ther i s o f ten ar ound".  This  j ust m eanst h at  he  i s often in t hehouse  w ithme. Heis often doing thi n gsin the house.  In  sen t e nc e ( 1 ), weca n  se e the same struct ure as inth e  e xamples ab o ve.  W e havethe ver b use d  and asubordina te clause :
She i s u sed to  [t here bei ng no one elsea ro und ] .
This m ea nsthatshe is use d tothe fa ct  thatThere  i s nobody  else around. In the ex amplewe se e t he --ingfor m  of  the verb  BE.  The subje ctofthisver b  is t he d u mm y p r onoun  the r e. It ist he  sa m e  w ord we seei n exi stent ial senten ceslike there is a n e w new presiden t. T h e  se nte n ce i sperfectly gram mat ic a l . "Nobod yelse around " meansth a t she  is th e only  person a ro u nd. Th ereisnobody ap art  f r om herin  t h at
 the_fact that_'Tom is in Paris'._Notice that_there_is a_non-finite_subordinate clause here.

Maria_is used to_[ Tom being in_Paris ].

In the_subordinate_clause, the verb is being. This is the  --ing form of the verb_BE._It is_non-finite_because_it does not have any_tense. It is not a_present or_past form of the verb. The Subject of_this_verb is Tom._Now look at the sentence below:

Maria is used to_[ being in Paris ].

Here the_subordinate clause also_has_the_--ing for of the_verb BE.  But there is_no subject for this verb. However,_we understand that the subject of the_verb BE is Maria. This is_because, if the Subject of_the verb_used and the Subject of_the subordinate clause_is the_same person or_thing, we do not need to_repeat it. We_can understand the sentence like this:

Mariai_is_used to [_____i_being_in Paris_]

We can see_that_there is_a_gap in this sentence, where we_expect_the subject of the phrase "being in_Paris" to be. The_identity_of this gap is_controlled by the subject of_the verb used.
In other words we_interpret the_sentence like_this:
Maria is used to [ Maria being in Paris ].
This just_means that Maria is used to_the fact that Maria_is in_Paris.
The_Original Poster's Question
Let's_look_at example_1:

She is used to there being no_one else_around.

Here the phrase be around just_means to be close,_or_to be nearby. So if I_live with my brother, I can_say that "my brother is_often_around"._This just means that he_is often in the house with_me. He is_often doing things in the house. In_sentence_(1), we can see the same_structure_as in the examples above. We_have_the_verb used and a subordinate_clause:

She is used to  [there_being no one else around ].

This means that she_is used to_the fact that There is_nobody_else_around. In the example we see the --ing form of_the verb_BE. The subject_of this verb is the dummy pronoun there. It is_the same word we see in existential_sentences like there is a new new president. The sentence is_perfectly grammatical. "Nobody else around" means that she_is the only person around. There is_nobody apart from her_in_that