Suppose I attack an X-wing equipped with Integrated Astromech with Rexler. I do some damage to the hull and have a focus token. Is the Rebel player able to use IA to remove a damage card after I flip the cards face up with Rexler's ability, or do they only have a window to do that before I use Rexler's ability?

Only beforehand. Flipping a card is not the same as dealing it.

For the same reason, Rexler's ability works normally on OT Chewbacca pilot.

IA works when a card is dealt. Rexler's ability works after after the card(s) lands one the ship which is after the "dealt" part of getting the card.

While it may not stop Rexler some have thought you need to use IA before you even know what the face-up cards you would be dealt are. This is wrong as when you use AI you should know what card you are using it on if it is public (face-up) information.