Quick Man's stage is, indeed, a b###h, and I can't beat it without the Flash (time-stopping) weapon. He's weak against it (a full charge will take away half his life) but I've always drained it!
One that always got me was Crash Man's stage, where you go up these long ladders while birds fly by shitting out eggs with chicks flying out, and not the good kind. I eventually just said #### it, held up regardless of the damage, and fought the boss with low health and a lot of rewind.
I played through MM2 not long ago on the Mega Man Legacy Collection with rewind, no save states, and got a shiny gold star on the game-select screen (Vol. 1 has the NES games, 1-6). I recently moved my Steam library to another computer and lost that and the save state at the start of Dr. Wily's castle I was working on in MM1. Majorly bummed about that.