Well honestly I think it would be helpful for him to speak to a counselor. The behaviors that you describe, the emotions he is experiencing are just not under rational control and are not appropriate to the situation.

I strongly urge you not to move in with him at this time.

Caring about someone, and caring about the relationship, includes facing the truth about the person and the relationship. And, when needed, it means asking for help to be arranged. If the person won't get help, then limiting contact with the person, or staying away altogether, can be necessary for self-protection.

When he gets really emotional about the so-called cheating I would just withdraw from the discussion. You know that you care so don't be swayed by his accusations that you don't.