Legal Question in Criminal Law in California

He is claiming malpractice (IAC) which vitiates the attorney client privilege. YEP, the attorney can reveal confidences and secrets to rebut the malpractice claim. Most attorneys do NOT do that. But these IAC appeals rarely go anywhere, best move on.

Your question isn't entirely clear. Your boyfriend doesn't need to file "a motion to appeal", since all convicted defendants have an absolute right to appeal as long as they do so in a timely manner.

You say that he brought a motion against his former lawyer, but that also is not how things really work. He may have brought some kind of motion alleging that the lawyer represented him badly, or he may be planning to sue the lawyer for malpractice. He may also be filing a habeas corpus petition on the grounds of ineffective assistance of counsel (IAC). Or he may simply be arguing in his appeal the trial counsel was ineffective.

Depending upon which of these options your boyfriend is pursuing and the surrounding facts, the lawyer may indeed have the right to disclose otherwise privileged communications in order to defend himself. That doesn't necessarily mean it's the right thing to do, but I don't know enough about the case to have an opinion on this point.