Меня for genitive and accusative.. Мне for dative and prepositional... And also.. Мной for instrumental Я as you already know nominative
In "remembers me", "me" is a direct object, so you use accusative меня.
In "gives me", "me" is the indirect object - there is an unspoken direct object which is whatever is being given, and English tends to drop "to" with indirect object pronouns like "me", as in "he gives [something] (to) me".
Here's a declension table of personal pronouns I wrote up which might help in figuring these things out: https://forum.duolingo.com/comment/29119997
Dropping "to" from indirect objects creates confusion in these kinds of sentences. They can sound like direct objects, but they aren't.
Only "Who remembers me?" is correct in standard parlance. With interrogative pronouns as the subject of the sentence there is no do-support required, probably because it's more important for the interrogative word to be at the beginning of the sentence rather than having something like "Does who remember me?". Rest assured this is no threat to "do-support" which is alive and well and keeping English out of step with every other language.