The "War" in Iraq

OK. Somebody explain it to me.

We went to Iraq because...al qaeda was there? As I understand it, that wasn't as true when the war started as it is now.

We went to Iraq because...Saddam had weapons of mass destruction? Well, that turned out not to be true. This sort of proves to me that "military intelligence" is a misnomer.

We went to Iraq because...Saddam is a bad guy? Well, that one is true but it wasn't the reason given when we first sent our young men and women there to be killed and maimed (physically and emotionally).

We went to Iraq because...we wanted to make life better for the Iraqis? As we have so often in the past, we tear down a country physically and politically, and promise to rebuild it later by supplying workers and money. In Iraq, however, the Iraqis are killing each other and killing the very American workers trying to help rebuild their economic infrastructure.