Does it come back up and irritate the stoma?
Nope, ideally, there should be no skin exposed between the wafer and the stoma anyway so there's no chance for stool to get on the skin whether you're awake or asleep. It's the skin you have to worry about - your stoma is part of your bowel - it's used to having stool on it :)
What about tossing and turning in bed?
Occasionally, excessive tossing and turning can loosen the seal on the bag. I daresay 99% ostomates have had an 'accident' while sleeping in bed either due to tossing and turning or due to the bag becoming excessively full. It's something you learn to deal with. You get up, have a shower, change to a new bag, change the sheets then go back to bed. As time goes on and you find the right system for you, these leaks will be few and far between (I've only had it happen to me once in the past 6 months).
Most ileostomates do tend to wake up with the Good Year Blimp on their belly. You just forgo the morning cuddle until you've had a chance to get up and empty your bag.
Do most of you wear a belt or something like that?
A belt is purely a matter of personal preference. I've never worn a belt but I know others who wouldn't go without one. I also know others who wear one for the first few hours after changing their bag to ensure they get a good seal then take the belt off. You'll work out what works best for you.
And lastly, do you have to empty in the middle of the night?
Usually I don't, but it really depends on what I've been eating, and how much. I know some people who like to set their alarm for a certain time so they can get up and empty their bag to try and stop it from getting too full (and consequently leaking while they're asleep). Others, like me, don't have to do that. I can eat up to an hour before I go to bed and sleep a full 8 hours before getting up and having to empty. Once again, it's an individual thing.
Keep those questions coming :)