Actually "handwritten paper" is fine, although in this case "paper" does not mean the physical material but rather a school essay or other written assignment. See definition 4 or 7 here. Example:
She handed in her paper late, but her teacher still gave her an A.
However it's not a phrase you hear much these days. See this Ngram as an example -- note the sharp decline since 2000. Students are more often required to submit electronic or printed documents (though I'm sure there are good reasons why a teacher might require a handwritten paper now and then.)
Of course, that's not your question. The reason I mention the other definition is because when you say:
Someone put a handwritten paper under my door"
The listener will assume you mean a school paper, written by hand. If you want to be clear that you're talking about a note, then that's what you should say:
Someone put this handwritten note under my door.
The listener will assume the note is written on paper unless otherwise specified:
Someone put this note, written in magic marker on the back of a cardboard pizza box, under my door.
Although a "note" implies there was a message on the paper intended for someone. If instead there was just some random writing that happened to be on a piece of paper:
Someone put this paper under my door, on which they'd handwritten something, but I have no idea what it says or who it's for.