No, that would not be a grammatically correct answer. Think of it as "Yes, I do (remember)" and you might notice "Yes, I am remember" doesn't make sense, as "remember" is not a state of being.

No, I am afraid that doesn't make any sense at all. The English speaker would immediately think - 'he is what....?'

Just an extra note: the phrase you need is "reply to". I see you've translated directly from Russian, and that's caused the error. We never say "reply on".

No, you cannot say "Sure, I am" in reply to "Do you remember?"