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Villain is an old lady probably in her late 50's. Haven't seen her bluff. My read on her as being solid. 

http://www.holdemhandhistory.com/play/7339/

What's my river play and did I play streets correctly?

I think the turn is a clear bet. As played, the river is a clear call. By playing the hand passively you've underrepped your hand, so she could be betting a Queen with a worse kicker. The majority of hands she's betting for value beat you, but you're getting 5 to 1 on the call.

If there were more draws I would think about betting the the turn reasonably big. As is, it's likely she has a Q or some sort of mid pair. There's some chance of pure air, too. What may be best is to bet like 1/5 on the turn and 1/3 on the river. She won't fold 2nd pair to such a small bet, and that's better than checking. It then allows you to get her to the river with those hands, and she may not fold to a 1/3 bet with 2nd pair on the river.

As played, river is an easy call.

You should be betting the turn for value. Target all of her Qx hands. The only way she has a K is if she has exactly KK, AK, or KQ. And it's unlikely she'd check any of those on the turn. Pot is $302 on the turn, I like a bet in the neighborhood of $100-$125 if you're trying to get 7x and hands like 88-JJ to call in addition to all of her Qx.

That makes the river ~$500 and you can pop another $200 bet.

Also, more info than "she seems solid" would be helpful. That's not much of a read; why does she seem solid? What have you seen her do?

Finally, we really need stack sizes here. If villain is shorter I'm setting up my bets to make a comfortable size on the turn to make an easy 1/2 pot shove on the river.

I agree with above; we need stack sizes and better reads.
I like a bet/fold on the turn and if called a bet on the river.
As played call the river.

Just a curious question. Wouldn't betting the turn be bad if the lady folds out all worse Queens? I mean, I guess we would be giving her a free 3 outer (and/or 5 outer if she has a hand like 87s) on the river if she has like QTs, but if she is folding QTs when we bet the turn, why not play our hand like 88-JJ here and let her bet river (and/or, we bet river)?

Checking the turn is fine if we are certain that we will gain more value by on the river than we are going to lose by giving away a free card. If, it's true that a free card is not likely to hurt us AND that we are going to get more more out of our villain on the river than we can on the turn, then check away.

I don't see why a Queen's folding the turn by calling the river, but if that's the read then go with it.

You have the best hand OTT like 100% of the time. Old people will not C/R you with worse to "get you off a range that is capped at 1pr"... they don't think that way. She will also be very unlikely to donk flop with better than AQ - she's really unlikely to have overpairs KK/AA and unlikely to have 2pr hands {Q7,Q2,72}. As for sets, she's probably not donking out with them on the flop... she'll check and let you c-bet then do one of the following:

i) check/raise
ii) check/call, check/raise
iii) check/call, check/call, donk huge.

I really don't see why you check the turn. The K is irrelevant, so bet for value.

Her range here is a lot of random pairs {33-66,88-JJ} and {7x, Qx}. I would say that she has QJ/QT here ALOT ALOT ALOT. Yeah, she could have KQ but since she checks the turn that is pretty much no longer a part of her range (here's why bing in position helps).

River sucks cuz she probably isn't value betting thin, so puke-fold.