When you offer a draw you also move :P. ... Just like 'real' OTB chess. If you choose to wait around ... lol.
Online Chess: clock keeps ticking while you wait for response to draw offer?

Yes, it's reasonable. Not replying at all is a perfectly normal way to deal with a draw offer. In the meantime, you still have a move to make, and when you do so the offer will still be active. Why would an opponent accept a draw without seeing your move first?
Otherwise you could make unreasonable draw offers (when you have a worse position) to gain time to think.
Over the board, this doesn't happen - you have to make your move, offer the draw, then press the clock. Here I usually make a move, "stay on this game", and make the offer right after I made my move.
In Online Chess, I've noticed that if you offer a draw to your opponent, your clock still keeps ticking down while you wait for his response to your offer.
Presumably this means you will lose the game on time if your opponent fails to respond to your offer? Does that seem reasonable?