Although it's not the proper time to offer a draw, if a player offers a draw before moving, is their opponent really not allowed to accept it until after they move? (I know their opponent can wait until they move before deciding to accept it, but do they have to wait?)
No, the players can agree on a draw at any time.
Although there was this kid (teenager) who kept offereing me draws every time it was my move for 4-5 moves in a row, I got up and told the TD, so the TD came over to watch -- the kid must have known it was wrong because he stopped without anyone needing to tell him.

Although it's not the proper time to offer a draw, if a player offers a draw before moving, is their opponent really not allowed to accept it until after they move? (I know their opponent can wait until they move before deciding to accept it, but do they have to wait?)