I have Delay the Enemy in play, which has the Battle trait(Characters use Attack instead of Willpower when questing). From what I gathered from the side quest rules, I can choose which quest I want to progress at the start of the questing phase. Does this mean I can announce Delay the Enemy and use my characters' attack to make progress on locations and avoid threat, even if I wouldn't be able to progress Delay the Enemy?

Yes, you'll choose Delay the Enemy at the very beginning of the quest phase, before you decide anything else. Then you'll quest as per usual, but because Delay the Enemy has the Battle keyword, you'll use Attack instead of Willpower. When placing progress tokens, you'll place progress tokens on the active location first (if any), then on Delay the Enemy. It doesn't matter if you weren't able to actually place progress tokens on Delay the Enemy (because you didn't have enough attack power, or whatever).

Thank you