Mexican Citizens on Death Row in the United States?

So there is a Mexican Citizen on Death Row in Texas (go TX Death Penalty Laws!!) who was a part of a gang rape and murder of two teenagers 15 years ago.

The UN and World Court is now trying to stop the execution.
The Vienna Convention is such a load of crap.

I have a couple questions...
1) Why has it taken so long for his execution to actually take place? He is illegal. He committed a crime on US soil, against an American. Therefore, in the eyes of most Americans, he can be considered a terrorist, yes? Immediate death. They wouldn't have any problem executing an American anywhere else in the world if something like this happened.
2) Don't you think actually following through on the execution of a CONVICTED killer/rapist might send a message to others who think they can come across our borders and commit crimes that we just might not put up with the BS anymore?
It might at least send a message to the voting citizens of this country that our political leaders will no longer be soft on crime, or on illegal immigration.
3) Why do we give a rats ass about what Mexico wants? They are a separate country. They are not taking care of their own people. Why should the US have to be their doormat?
and finally...
4) Why is the US listening to the UN? No, seriously. Someone explain this to me. And then tell me the last time the UN did anything worth taking seriously.