Mostly. "He seems happy." / "He appears happy". In these cases (which I agree happen often), you can swap them over. However... "A faint image began to appear." (correct) "A faint image began to seem." (nonsense) So I suppose that "seem" needs to be followed by a description. "Appear" can stand alone.

"Appear" could mean "occur" (opposite to "disappear"), aside from that, in your case, it has the similar meaning of "seem". Yes. Mostly the same. But if you're in China, as you actually are, you need to pay attention to what the teachers have to say: seem like...; but "appear like" would be incorrect. There's still this argument --- "seem + subjunctive" (it seems/ed as if/though he were/was Batman); whereas "appear + subjunctive" is considered incorrect/a rare collocation in some grammar books.

Yes, you are right.