Gee do US nationals get access to diplomatic authorities if they are arrested for capital crimes in other countries? I would bet there are plenty of armpit nations that wouldn’t extend this courtesy to US citizens... treaty or no treaty. Besides this is solely a state matter. The guy committed murder in Texas and was given a presumably fair trial in state courts and likely spent years in appeals before they finally are getting around to executing him. Obama should butt out.
Generally yes, but you have to ask for it. Nobody is under any responsibility to tell you about rights you may have under international treaty. You are expected to know them before you travel. Even then, the consulate generally tells you that you shouldn't have broken local laws, you're on your own. The government doesn't just pay for a local lawyer for you. Theoretically they might help you if you were charged for something that's not considered a crime in America, and you were ignorant of it being a crime there, and it became a political thing. But this is rape and murder, no excuse of ignorance applies.
OTOH, he didn't travel of his own accord, he was brought here as a child. Still, he knew his illegal status, so he should have gotten educated about how treaties could affect his status here.