Do you not get the first win of the day bonus if you start a match just before the quest refreshes

NO. This annoys me. You have to have the quest before you start. AND, at least for me, when the 22 hours rolls over, I have to relog for it to actually re-give it to me.

I don't think so. It seemed like it happened to me a couple days ago. Thought I might have started when it had like 17 min left and the game ended at 15, but it may have just not started while in the game.

It is now an actual quest, and yes you will not receive credit for it if you start the match before the quest is listed.