"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.