You are right, they will chose one chain. But that doesn't say that they can check incoming deposits against the deposit on the other chain. When they received the bitcoins not only on their chosen chain but on the fork too then they can accept that deposit because it is riskless.
Why should they not monitor the other chain and instead close down completely? That makes no sense businesswise.
You are talking nonsense (again) - if you accept coins that end up being on a failed fork then you cannot spend them. Got it?
So if I am exchange A and I accept coins on Fork A and am giving someone fiat for those coins then I've already spent the fiat and end up becoming a bag-holder of a useless alt.
Now tell me why any business is going to take on such a risk?