Call river with 44 (on board), Ace Kicker v. strength in 9.75 BB pot?

The continuation bet on the flop was marginal. It's almost 100% likely someone has caught a part of that flop. JT and two spades hits a large variety of hands people call raises with.

Give it up on the turn. At best you're behind QJ and have your AK for outs. Mostly likely you're behind AJ or KJ. Plus, one of your gutshot outs could be tainted if someone's drawing to a flush. I'd say you should assume your draw has 5 outs AT BEST. That's not enough to call the turn. Your gutshot is the only one with good implied odds. Hitting the A or K is more likely to have reverse implied odds.

And definitely fold that river. Your ace-high is so unlikely to be good here. I know you've read folding the winning hand is a HUGE mistake and all that... but think about the way the hand played out. Does it make sense for him to continue betting and betting here without being paired? He could have checked behind on the turn or the river, but he kept firing.

If your A or K paired up on the river, then of course call it down. You have some chance of winning there. And if ANY Q came on the river, assume you're ahead and raise. It's unlikely he had a flush draw here.