I believe that paying by gift card will count. I think there's a pretty good opportunity to triple dip on this deal, buying a gift card via the UA MileagePlusX app and then using that to pay for the purchase.

As for the timing, Amazon reports numbers to affiliate partners on a monthly basis. I receive mine 60 days after the close of the month where the activity took place (e.g. on December 1st I received data for September 2016). JetBlue would then have to process that data and credit the points. I would not expect the timing to be too much faster for them.

As for the 24-hour window to complete a transaction, that's also tied to the way Amazon runs its affiliate program. Make sure you're not using an ad-blocking utility that blocks tracking cookies or you can also expect to not receive the points.