You need a clear line of thinking.
It's very hard to gauge the villains betting, because he seems to bet very small. And this can be interpreted a few ways - so my thought would be, how has his previous betting looked? Does he consistently make small bets? If so, then I think we put him on a standard value raise: AA, KK, AK, maybe QQ. If your opponent generally uses more traditional bet sizing, then I think we remove AA and KK from his range, and his range looks more like AK, AQ, QQ, JJ, TT - hands with value where your opponent wants to take control of the hand without having to spend a lot to do it. And it's very important to puzzle this out pre flop because the hand plays very differently depending on which assumption you use.
Standard TAG with small bet sizing: when the K hits on the flop I think calling isn't a good option. All of his range is now ahead. The question now is: do we figure out a bluffing strategy, or fold? I think given his perceived range of AA, KK, AK, and maybe QQ, it's unlikely we fold out better. We can try to rep Aces and assume he has AK... given that his bet is on the small side I think raising Aces would be believable... if the player is a thinking player. But, getting people to fold top pair on micros is no easy feat.
I think fold flop is the optimal play.
Unless we think our player 4 bet light in an attempt to take control of the hand.
Now we think your Queens can be good right? So what hands lead out that you can beat? How about AQ of hearts? Maybe that TT & JJ I keep talking about. So right now, his range is AK ahead, with 12 combos; and TT, JJ, QQ, AQ hearts that you beat (or chop) - 14 combos. Given that we're roughly even with his range and getting a good price, a call is fine.
Now, another King comes out. This is VERY IMPORTANT. There are now 4 less combos of AK that beat you! So if we called the flop, we are actually in better shape on the turn. Especially considering his bet is so small, I don't think we should ever fold the turn. Honestly, my thought would be: is there any way I could raise the turn and get called by JJ, TT, or AQ hearts? I think given our large pre flop raise, unfortunately probably not. Maybe we can get the one combo of AQhearts to stick around....
When the river blanks out these combos become the crux of our decision. If we think we may be good on the flop, and looking better on the turn, there's no reason to not believe our hand isn't good on the river. We are essentially split in our combos, 8:14. We are basically going to be ahead here 66% of the time, ish. With getting 3:1 on our money, we only need to be ahead 25% of the time, which makes this a snap call.
To reiterate: if we feel we are likely enough good on the flop, we are more likely good on the turn, and getting a good enough price to call the brick river.
So I think we only have two lines: fold flop, or call, call, call. And we know which line is best based on our history of watching our villain's bet sizing in previous hands.
It is important that call, fold; and, call, call, fold are NOT good lines.