Ace High vs draw...

I encountered a situtaion, where, had I been too results oriented I might love. But I soon realized that perhaps my play was spewy/-EV.
I was mid position, folds to me I had :Ad::Jd:
I raised about 4xbb. one caller, the BB
Flop :Qh:
BB checks. Now I had him probably on connectors maybe suited maybe not. Q no kicker, probably would've folded, Q Good kicker, probably would've raised(this guy anyway, Pretty TAGG I would say) So I bet into him about 2/3's pot. Call
Turn:
BB checks, I bet a little over 1/2 pot. call. Now this point, I was going to shut down, cut my losses.
River: :Kh:
Now I see that if infact had the hand I put him on originally, he missed any kind of draw. And after his check, it was decided. I bet about half pot. BB folds.

In the words of the Late Great Lynyrd Skynyrd "Was I right or Wrong"

Follow up bet is good.
If the villain tends to call follow up bets no matter what, then I do not suggest you bet at all if you don't have a hand. I would check this behind.

Turn: If he called the flop he will call such a weak bet now pretty much every time. Many players will call a 3/4 pot bet on the flop and will fold to a 3/4 pot bet on the turn. - but this isn't such a horrible play if you hit a piece of the flop. Most of the time in a cash game both players are pretty deep and the initial raiser might just make a follow up bet and TPNK might be good. If he does have a good hand, you can still fill up and you get plenty of implied odds if you hit trips or 2 pair on the turn. I'm not saying this is right - but you can't expect to get a player to fold on the turn who called on the flop if you give him 3:1 on a call on the turn.

On the river, if you think you have the best hand, why on earth would you not just check behind!!
What do you want to accomplish? Are you value betting, hoping to get a call from K high? Do you really think you can get him to lay down a pair of Queens, 8's with another 1/2 pot now?

The title sais "top pair vs draw". You realize that any type of draw is more likely to make a hand than it is for your A high to hold up?
Count the outs! There's a good exercise in pot odds right there!

First of all, no one is going to fold Q no kicker to one bet.

Checking behind the flop and cbetting are both OK though. The turn is kind of spewy because he rarely folds what he peeled with on the flop.

I guess once you bet the turn you have to follow up on the scary river.