Piece of paper should be accepted. I rarely say "sheet" even though it's likely more correct.
Neither is more or less correct but "sheet" of paper is more precise since a piece of paper could be anything from a fresh new sheet to a torn half to a scrap in your pocket.
Maybe it depends on dialect becuase i say sheet of paper
Yeah, as a Korean person myself this sentence would NEVER be used. One would say "이 종이를 갓고 모하라는거야?" (Though it sounds rude..)
'What shall I do ...' should also be accepted?
I wrote "What will I use this sheet of paper for?" Exactly the same meaning so it should be accepted.
Spliff 굴리다 ;)