You're right that your hand is quite under-represented, and you certainly shouldn't fold it getting 3:1 when people could easily value bet worse. The real problem, I would say, is the earlier checks. Also I'd size a bit larger pre-flop because the original raise was quite small.

Flop is a bit close, but I think betting is better than checking. Would you ever bet here without a pair?

Turn is a very clear bet. Just because better hands are possible doesn't mean you should check. There's quite a bit of value in being called by worse Ks, worse pairs, and heart draws, and even in folding out weak single-heart hands like 8h 8c that actually have a better than 20% chance of outdrawing you on the river if you let them see it for free.

Failing all of that, you should value bet the river yourself rather than check and call. The only merit of checking and calling would be to induce a bluff. Otherwise, although you're right that people may value bet worse, they will surely call with more worse hands than they'll bet.

I encourage you to watch or rewatch my Getting Paid series.

I'd bet again on a blank river, and fold to a raise. You're not going to be able to avoid tough spots in poker. You just have to learn how to deal with them. Keeping your ranges balanced is the best way to cope with unknown opponents who will bluff with an unknown frequency. This is part of why I don't like checking the flop. You seem to realize that it turns your hand more or less face-up, and this is the reason why you feel like you can't put money in good even when you turn a really strong hand. You'll probably face the same problem if you have QQ in this spot, bet the flop, and get called. Since you aren't betting AK, are you ever betting without a pair? If not, then QQ is actually one of the weakest hands you could have, even though it's pretty strong. All of your discomfort here is stemming from being unbalanced without any clear plan of what you're trying to exploit.