It sounds like this is all very recent. You say he has been in therapy twice a month for 5 months - has he been with you guys only 5 months? That certainly seems implied by his being 8 years old and having been with his mom during second grade. He's probably just started 3rd grade in August or September I would imagine. In terms of therapy for understanding and healing from almost 8 years of instability and abuse, 5 months is nothing. And as an aside, is there a reason he doesn't go to therapy every week? Two weeks feels like forever to most kids - he may not be able to grasp therapy as an ongoing process with so much time in between.
It sounds like you guys are doing half of the equation really well: the consistency half. It's fantastic that you guys are providing a stable environment with predictable and reliable expectations. This is absolutely key for all children, but especially those coming from unsafe/abusive situations.
However, it also sounds like there's a huge focus on his behavior as wrong and in need of correction. As in, "we need to fix him because he is a damaged person as a result of his experience with his mom, and we'll know he's been fixed when his behavior conforms to such-and-such an image of what a prototypical 'good' child's behavior should be."
If his mom was, as you say, such an alcoholic, the child is likely used to walking on eggshells at home, because it's very dangerous to upset an alcoholic. It makes sense that he would be on best behavior at home. Oddly, he may actually be acting out at school because he feels safer and more in control there. Someone asked about his behavior at his prior school - did he act out there, too?
The half of the equation that seems to be missing (which may be more a result of the focus of your original post rather than a reflection of reality) is affirmation, compassion, and encouragement. When he comes home with one of these check marks, do you start with, "why did this happen? Don't you know it's wrong to...?" Because the fact is he might not know. He might be reacting to emotionally difficult moments in the only way that has been consistently demonstrated to him, and therefore the only way that he understands, and that makes sense to him.
My recommendation: start by listening to his description of the incidents (rather than going by whatever description was sent home with him - that description will probably give you the most accurate dry facts, but you need to know what it looked like from his perspective- including his feelings, and what he perceived others' feelings and intentions to be). Then empathize with him - "I understand why this made you feel that way, and I'm sorry that happened." Then, if you think based on the information you have that his interpretation of the situation was off-base, explain: "I don't think Generic Boy was ignoring you on purpose. It's more likely that he was just playing with the friends he knows from last year. It always takes time to make friends at a new school, especially when the other kids all know each other already." Then address the reaction: again, start with empathy - "I understand why feeling hurt and rejected would make you want to xyz, but I think we can come up with a better way to handle this kind of situation in the future." Then I would come at the solution from two perspectives: 1) Results. Probably there is something he wants to achieve, such as eventually making friends with people at school. You can ask, "did your response help you with this goal?" Likely the answer is a resounding no. Help him come up with a way he could have responded that would serve the eventual goal of making friends (or whatever it happens to be that he wants to achieve). 2) Social conventions for what is appropriate/acceptable behavior. Because his mom did not demonstrate this aspect to him, it should not be taken for granted that he knows. Explain to him what is generally regarded as appropriate behavioral parameters in the given situation and why. This last may be hard, because once a concept of good and bad behaviors is instilled in us on a more or less instinctive level, we sort of stop thinking about why. But the reasons will be very important for him to understand if you want appropriate behaviors to become internalized for him at some point.
Last: take the time to observe and affirm when he does things well - with particular focus on handling emotions appropriately. If he's getting worked up in a conversation, for example, and when you point it out he actually calms himself down, let him know how impressed you are with his ability to handle those difficult emotions so well. You can help him recognize other areas where he does well, too, so he can start to build a better sense of self-worth - because ultimately acting out tends to be a response to insecurity caused by feeling inadequate or unsafe. The safety will come over time with the consistency you are providing, especially if home is an emotionally safe environment. Self-worth might take a little more work, especially if the focus is always on what he's doing wrong. Start focusing on what he does right.
