Question 2: Yes, when a permanent is exiled and then returned to the battlefield (flicker) it is a new permanent, not the same permanent (as per the rules) and will therefore enter under its owner's control
from ixalan's binding card:
exile target nonland permanent an opponent controls until Ixalan's
Binding leaves the battlefield.
So first of all simply put, when you demystify ixalan's binding this part would resolve, so the Immortal Sun would return to the battlefield. I get the question might be why it comes back to you and not your opponent so...
Rule 108.3 states that the owner of a card in the game is the player who started the game with it in his or her deck. The card would return to its owners control...it couldn't come back to its "controller" control because at that moment there was none.
You own Immortal Sun, your opponent casts it and gains control of the card, but not ownership, you exile and now the card is not controller by anyone anymore (after the trigger resolves, before it resolves it is under their control or else ixalan's binding wouldn't be able to target it) after it returns from exile it is now a new permanent that must enter under its owners control as there was no controller, it then becomes a permanent when it actually enters the battlefield and you become the controller.
Hope that sort of makes sense.