Catalyst Field third effect
I will answer your questions in order,

1) At what point does my opponent's card get destroyed? When I banish the Gemini monster or when it returns at the end phase of my opponent's turn?

It will be destroyed after you banish your card as the banish until your opponents end phase is a condition applied to the effect, and then the destruction happens in the effect resolution.

2) If my monster is banished by this effect and my opponent destroys Catalyst Field while the monster is banished, will it come back at the end phase or will it stay banished?

If Catalyst Field is destroyed while your monster is banished yes it will still come back providing there is an available zone for it to return to as I stated above it is a condition of the effect. This is similar to PSY-FrameLord Omega who banishes himself and an opponents card, providing there is an available zone for him to return to he will return.


  Once per turn, during the Main Phase (Quick Effect): You can banish
  both this face-up card from the field and 1 random card from your
  opponent's hand, face-up, until your next Standby Phase. Once per
  turn, during your opponent's Standby Phase: You can target 1 banished
  card; return it to the GY. If this card is in your GY: You can target
  1 other card in the GY; shuffle both that card and this card from the
  GY into the Deck.


3) The banishing of the gemini monster is a cost? For instance, if my opponent activates something that would negate the activation of a spell effect (not card) in response to the banishing effect of catalyst field, would my monster get banished anyway (as a cost)? Or would negating the activation of the effect stop the banishing of my monster altogether?

Banishing a gemini is not cost for this effect, in yu-gi-oh cost is indicated by a semi colon (;) so in this instance the cost is to target an opponents monster, then if the effect is allowed you will be able to banish your monster and destroy your opponent's. Cost in yu-gi-oh cannot be negated as it happens before the effect attempts to activate it is what needs to happen first. Indicated by a popular scenario in the current game with Pot of Desires and Ash Blossom &amp; Joyous Spring
 .

Pot of Desires.


  Banish 10 cards from the top of your Deck, face-down; draw 2 cards.
  You can only activate 1 "Pot of Desires" per turn.


Ash Blossom


  When a card or effect is activated that includes any of these effects
  (Quick Effect): You can discard this card; negate that effect.  
  
  
  Add a card from the Deck to the hand.  
  Special Summon from the Deck.  
  Send a card from the Deck to the GY. You can only use this effect of "Ash Blossom &amp; Joyous Spring" once per turn.
  


Pot of desires can be activated and then the player performs to action up to the semicolon which in this case is Banish 10 cards from the top of your Deck, face-down; then your opponent has the opportunity to respond where they will discard Ash Blossom which negates the rest of the effect to draw two cards, so in this scenario the turn player has lost 10 cards from their deck to gain no cards back as the effect was negated.