Crusoe:
First off poor old you - I'm sure you're trying your best and clearly want the best for your DS.
Adoption is tricky and although you may know quite a lot about his background, you may not know everything nor totally understand why he has this pattern of behaviour (even if you do recognise that change triggers it).
My advice is openly discuss with the school whether an educational psychologist might help.
If all this poor lad sees is you being grumpy with him, disciplining him on the say so of other people and perhaps feeling worried to tell you things because you will be upset that will harm your relationship. If all he gets from school is punishments - (and possibly labeling him as a trouble maker) - it kind of creates a negative feedback loop. From his perspective he may just see this as 'nobody likes me' or 'nobody sees the good in me'.
I know it may be hard but my advice is to avoid disciplining him for things when you're not around (unless absolutely major - i.e. injuring another child) - but try and discuss it. Quietly, calmly but also in terms of exploring how other people in the situation might be feeling.
The main problem may be that at this young age he really is only focused on his feelings and they may be quite forceful (so immediately getting angry, or if embarrassed showing off, or doing something rude to get attention). The burping does strike me as attention seeking behaviour - and at core what he may be after is attention from these new people in his life and he's made some bad choices about the type of attention he's attracting (which sounds largely negative).
So maybe a quiet chat with him about how he'd like teachers to see him and behave toward him and then getting him to see that his behaviour can help achieve that outcome. I suspect he wants to please them and he wants to do well - he's just not completely clear how to do that and (like most boys) he may be so full of beans that sitting still & behaving for more than 10 minutes is asking quite a lot right now. If excess energy is a factor - seriously consider getting him more involved in sports.
Finally - do try and remember that at this age you may not be getting the whole story or a clear version of his perspective:
My DD2 was constantly getting in trouble at nursery for scratching people. She's got incredibly strong nails and if I don't keep on top of them and cut them frequently they will scratch. Now it turned out she scratched a girl on her face. She was punished at the nursery and the mother went postal - accosting my DH in the car park and screaming your child has ruined my beautiful daughter's face, etc.... before even introducing herself. The nursery had punished my DD2 immediately and DD2 was inconsolable - she was so upset. About 2 years later - I saw her with her cousin making scratches on each others arms at the end of a summer holiday when we were all heading our separate ways. I asked what are you guys doing? 'We're make a scratch so that we always remember each other,' was the reply. On the trip home I asked DD2 if this is what happened at nursery - she said yes, the girl concerned had asked her to do that so she would remember her when she went to big school. I felt awful. Poor mite - we all assumed she'd been the trouble maker and she wasn't. So my one word of warning is don't rush to judge your little boy - hear what he has to say - give him time (because it may be difficult to express himself now) to explain what he's feeling and what is upsetting him/ setting him off or who is encouraging him to behave this way.
With DD2 who is very easily upset and not happily settled at school - we've adopted an Oh well, tomorrow's another day approach and it has really helped. We discuss what happened that day. We agree what to do better next time and what (or sometimes who) to avoid and then we move on. I think it helps her believe that we think she's a good kid and it was just a 'blip'.
HTH