AQ may dominate KQ, but it doesn't always win

$10/20 limit game with a 1/2 kill, this was a killed pot, so the action was $15/30.
I was in the small blind; the kill poster checked his option, everyone else folded to a middle position player who raised. A late position player called.
I peeped AQ, and felt the raiser was pushing a mediocre hand.
I three-bet, the big blind -- a loose passive player who almost never folded before the flop -- called, as did everyone else.
The flop hit A, Q, J, rainbow.
I led into the pot -- which was swelled with $225 in pre-flop calls -- with my top two pair and everyone called.
The turn was a ten, putting four parts of a straight on board.
I checked, and it was checked to the initial raiser, who bet.
It was folded to me and the big blind was holding chips to call.
I slipped my $30 in to call, knowing I need an A or Q to win the pot, and the big blind did likewise.
The river was a blank, I checked, the big blind checked and the initial pre-flop raiser bet.
I wish I was typing that I made a smart fold and saved a bet, but I called, the big blind folded, and my opponent showed KQ -- a straight.
While my AQ dominated her KQ, she still made a straight and took the pot, and I was still a donkey for paying off the river. On the turn -- getting pot odds of 330 to 30 [11 to 1], and anticipating the big blind's call, which would make the odds 360 to 30 [12 to 1] -- I was getting close to the right price to chase my four out draw [11 to 1 against].
But paying off the river -- dumb mistake.

Side note: my opponent made an aggressive play with KQ pre-flop, but not necessarily a mistake. When I re-raised, she had proper odds to call to see the flop.
On the flop, she had middle pair and a gutshot draw, so her call was smart (calling $15 with $270 in the pot and only one player -- who had only passively called to that point in the hand -- to act behind her). A gutshot is about 11 to 1 against to hit (same odds I later had on the turn when I needed an ace or a queen to make a full house), and the pot was offering odds 18 to 1.
Of course, this gets more complicated if she factors in the likelihood of my holding AK, or another hand containing a king and whether or not the big blind has a king. If either of us holds a king, it would be a split pot, changing the pot odds. If we all have a king, the pot could be split three-ways, changing the pot odds even more!

*******If any of the odds or lingo above is confusing, you need
DRAWING DEAD TO A GUTSHOT: How to Talk Poker (http://www.amazon.com/gp/product/1569803080/ref=sr_11_1/102-7480320-0644163?ie=UTF8)
and Poker Flash Cards for Texas Hold'Em: Learn the Odds to
(http://www.amazon.com/gp/product/097650250X/sr=1-1/qid=1156289481/ref=sr_1_1/102-7480320-0644163?ie=UTF8&s=books).