I am not sure I have ever said "sheet" without "of paper," unless I was talking about beds or sailboats.
Yes. Well, I don't even know what else could be described as "kartka". You have "kartka/karta pocztowa" (a postcard), sometimes "kartka" is enough to describe it, if the context makes it clear. And you had "kartki" in the communist times, which allowed you to buy sugar, meat and stuff like that in a given quantity. That's all, I think.
Isn't "a paper" an equivalent of "a newspaper"? Or in more colloquial language, some document?
When someone says "give me a paper and a pen," the reasonable supposition is that the person needs a sheet of paper and a pen with which to write. It is not asking for a newspaper. Of course, a "paper" can be short form of newspaper, but not in this context.
Sorry, but no it isn't. My supposition would be that you wanted a newspaper and that you wanted to do the crossword, or perhaps mark some ads.
The paper you write on is uncountable/non-count. I've never heard anyone use "a paper" with this meaning. It would be "a piece/sheet of paper. "A paper" would be a newspaper or an academic text of some type.