In chess, may a player resign, offer a draw or claim a draw at any time, or not?
You may resign at any point during the game, there are no restrictions at all.

Draws are regulated in article 9 of the FIDE Laws of Chess:

You offer a draw after making your move, before you press the clock (9.1.2.1). The offer stands until the opponent touches a piece to make his move (or he accepts or declines or the game ends some other way).

If you offer it at another time, the offer still stands, but it may be counted as disturbing your opponent and you could get a warning from the arbiter. Besides that, your opponent will just not react to the offer, wait for you to make a move and then make his decision.

So in particular in your case 1), you do not have the option of offering a draw then -- you need to make a move first.

You claim a draw on your own move only, before you touch a piece to move it. Once you do that, you have to move and can't claim anymore (9.2.1 and 9.4).