I think I'd cbet this flop for both value and protection.  If you check back, there's a lot of turns that he can (correctly) lead and put you in a tough situation.  Non-spade 9/8/6/4/3 are bad, diamonds are bad, and even J might hurt us; even if we flat on these turns, it's gonna be tricky if he fires another on the river.  On the other hand you can have more KK than he, and he probably won't x/r this board as a bluff very often, so when I get raised I would probably fold.

As played jamming the turn looks kind of suspicious, as you're not repping strong hands very well.. you probably would bet wraps like 9876 on the flop, as well as sets and 2ps, and even hands like KQJ9 you might not always raise the turn unless you have FD to go with.  That said I expect him to hero call us quite often.  I usually bet this flop so hard to answer what I would do on turn, but I think I'd just flat and see the river and find some spots for hero calling or bluffing.

Not sure about flop, but I think checking is reasonable.  A decent portion of his range is smashing this flop(30-35%), and the stuff your folding out often has <20% vs you(which is another ~30%).

Not sure if jamming turn or calling turn is higher EV.  Your turn jam seems +EV if your getting like 20-25% folds and ~35-40% equity when you get it in.  Whether or not that's better than a call, I'm not sure.  I'm leaning towards a jam at the moment but I'm open to arguments for a call even if they're just range considerations.