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I am heads up against a player who is very aggressive. He raises almost every pot preflop and will bet out if he has anything, even 3rd pair

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So far seems like a standard for HU play. Aggressive play wins.

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I'm in the small blind/button with K7 hearts and raised frankly because I knew he would raise if I didn't.

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If this is really the only reason you raised perhaps you need some help with your HU play. Any king is good to start with HU, and I'm even raising trash hands in this spot so that I get action on my big hands.

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I made it 500 and he calls.

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Yuck, give him odds to call with almost any 2 here, eh? Raise more preflop, make him define his hand a little more - unless he's the type to completely disregard things like pot odds and make -EV calls.

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Now, he is a loose caller but he wouldn't just call any 2. He would call with SOME kind of hand, though, Even, say, 45 suited. By the way, if he had an ace or a pair, he would've reraised. OK, so the pot is 1K I now have 5700 and he has 4300.

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Oh but you gave him the right odds to call with lots of hands! That's the point here, he only needs about 40% equity here to make a call right. HU that includes lots of hands.

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Flop comes 10,7,2 all spades. He checks to me. What is the right play?

Things to consider:
1. He is very aggressive, so the fact that he checked means he seriously doesn't have anything. If he had a pocket or ace high he wouldve reraised me preflop. If he hit any pair on the flop, 2 pair, or even a set, he would've bet out even though I was the preflop raiser. So I was pretty sure that I was ahead. The only thing he has that currently beats me is if he flopped a flush.

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Monsters under the bed, huh? Bet already and see where you're at. You make your own case for it. Grow the nuts and put the chips out.

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2. I can't give him a free card because pretty much whatever comes on the turn is a scare card. Unless it's a 3,4, or 5 non spade, he could've improved. And if I check, he is guaranteed to bet the turn whatever he has, and I'll have no idea where I stand.

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Ok so you've made your own case in favor of betting. Remember that if villain isn't holding the hand you're playing scared of that he's thinking you could have that same hand. What's the problem here?

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3. Because he is very aggressive, he is capable of coming over the top of me if I make any bet, and so if I bet, say, the pot which is 1k, he could easily push on me and I won't be sure what to do. I'd probably fold.

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He could come over the top of any bet in any hand. Are you going to sit there until you have the nuts before putting another chip in the pot?

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4. I have 5700, he has 4300, pot is 1k. A push feels a little extreme. But if I bet 1500 or less, I know he'll push with any 2. What should I do???

Results to come later

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You already made your own case in favor or betting or pushing yourself. This isn't that tough of a decision, especially with a good read on your opponent like you had. If you really think you're ahead here then take the gamble and put out a bet. If he does push you're getting the right odds to call and it's not even close.

If you're not willing to play for villains whole stack you could always just check and give the pot to villain on his turn bet and give him the chiplead. That's what will happen if you don't bet and you know it.