Case (1): Let me rewrite it to:
Maybe he's right. Isn't he?
This is the most usual way a native speaker would go with. The or feels redundant in your original sentence. You should keep it out.
Case (2): It could work, but not as a question, rather as a hesitation:
Maybe he's right. (thinking) Or he isn't ... (dramatically)
Cases (3) and (4) both sound clumsy and strange. The biggest problem in your examples is that you're missing it. The proper way would be:
Maybe he's right. Would it be a mistake to even doubt what he says?
You don't usually "doubt someone's falsehood." I can't imagine what that would do to the universe. You usually "doubt someone's opinion / statement."