Why it's wrong:
I carelessly assumed (without, of course, noticing that I was doing so) that when a line is completed it's always at the bottom of the board. Of course that is very much not necessarily true. This invalidates everything I said about #2 and #3.
Wrong answer preserved below because I don't believe in hiding my stupid mistakes:
Two of the answers are
no, to "Level 1" and "Level 3",
if you imagine colouring a 7x4 block (in whatever orientation you please, whatever the total number of columns) in a checkerboard pattern, clearly it will contain equal numbers of black and white squares; six of the seven Tetris pieces also contain equal numbers of black and white squares however they are placed. This immediately shows that #1 is impossible. #3 is also impossible even though lines may be removed one at a time, because each line-removal removes the same number of black and white squares. So if we could do this then there would be 7+4 events (7 "place piece" and 4 "remove line"), all but one of which preserve the white/black balance, starting and ending with white=black=0, which is impossible.
As for the third,
since 7 is an odd number it might seem that with 7 columns we can arrange some sort of colour-imbalancing shenanigans as rows are removed. Not so. Suppose that instead of removing rows we merely allow them to move downward off the edge of the board, and suppose that the colouring of a piece or piece-fragment moves with the piece. Then clearing four rows on a 7-wide board still just means forming a 7x4 block, and that we can't do no matter how we place the pieces.
None of the above is changed
if we give Joe the ability to place the pieces wherever he chooses -- not merely "sliding and rotating underneath" but also leaving them hanging in mid-air. So long as he can't actually place the pieces so that they overlap one another, the checkerboard colouring guarantees that he can't perform any of the feats asked for here.