Can I {Quicken} a {Time Reversal} before my friend exiles my graveyard?
Also, do I draw the card before I use the reversal or after?
Yes as long as you cast it while the exile graveyard spell is on the stack.
The order of how things go down is as follows.
Opponent has priority, casts spell to exile graveyard.
You get priority (an opportunity to respond before his spell resolves), you cast {quicken}.
Your opponent gets an opportunity to respond to you playing quicken before it resolves. He does nothing, quicken resolves, you draw a card.
With the exile graveyard spell on the stack, your opponent says I'm not going to play anything else, you get an opportunity to respond again. You play your {time reversal} spell.
Opponent gets an opportunity to respond again. No response, spell resolves, everything is shuffled.
Then the only thing still left on the stack is his exile graveyard spell. It resolves.