So I'm stuck in a quest to battle an opponent I can not win against. I've already used my skip, or whatever it's called, for the day and now I don't know if it's possible to quit the quest, probably not since there's a skipping option? What I do want to know is when the battle quests are reset at midnight will it close out the challenge that's just sitting there or am I really stuck?

Once you finish with your "skip" option, you can't cancel and can't move on to the next level until you beat the opponent. Tomorrow the BQ will restart to level 1 :)

there's no time limit on BQs anymore. Since you're stuck until tomorrow anyways, you could ask for advice in the Battle Forum, or maybe even borrow equipment on that thread over there.