[TheRempel]

Well even in hypothetical terms the button limp is bad.

At any given point in a hand you should be looking to create a situation where you can a) protect your hand and b) maximize the money you will make on later streets.

In this case you don't really need to worry about protecting your hand so much. An ace or queen on the turn or river will give you the nuts and second nuts respectively. A jack is ugly but you want AJ, QQ, QJ and JJ calling the flop bet without odds.

As for the second part, a large part of the reason you want to be here is so you can bet more on the turn and value bet larger on the river. If you check the flop and the turn is a brick your turn bet will be substantially smaller than if you had bet the flop.

The only reason to check here is if a two pair hand will fold the flop if you bet but call the turn if you check, and this is greatly outweighed by the other factors. It is generally correct to bet even when your opponent is correct to call and is getting odds.