According to Wikipedia it looks like we have spent a bit over 100 billion on all missile defense since the 80's star wars program. I imagine that the ABL only was a small fraction of this.

Actually, the American Polaris, Trident and Poseidon SLBMs are all two or three stage solid fueled. Titan II is not an SLBM but is indeed liquid fueled.

The ABL is meant to loiter near the launch site just outside of enemy airspace and engage when a launch is detected. During this time, you are right, there will be a massive heat signature. So the heat and the proximity to the target allow the ABL to track using radar and infrared. But remember that this is just a single layer of a multi-layered missile defense system. The "ballistic" phase you described is meant to be countered by the GMD (Ground based mid-course defense), which includes the EKV (Exo-atmospheric kill vehicle). And you are right that a warhead is significantly harder to hit at this stage.

In order to mount a weapons grade laser on a satellite, we would need to withdraw from the outer space treaty. This would probably be a bad idea currently as it would cause a rush to weapon-ize space by the US, Russia, and China.