The U.S. and the U.K. also have an extradition treaty, so the existence of a U.S.-Sweden extradition treaty doesn't really do much to enhance the plausibility of a double-extradition plot where Sweden extradites him from the U.K. on rape charges only as a ruse to get him into Sweden where he can be extradited to the U.S. on some other charges. (If the U.K. was a country that had no extradition treaty with the U.S. and was particularly resistant to U.S. extradition, then a double-extradition plot might be plausible.)
Now the U.S. and Ecuador also have a treaty but I don't read lawyer well enough, especially not old-timey lawyer, to know whether it would apply to anything Assange might be charged with. It does say it "shall not be applicable to crimes or offences of a political character" and I'm sure that's how Assange would characterize it (and I think that's at least partially why Ecuador has been housing him).
But the U.S. hasn't charged him with anything so there's nothing for Ecuador to extradite him for yet. In any case, the claim is a nice excuse to avoid jail in Sweden.
No, he isn't. Contrary to the ever-lasting myth, embassies have no extraterritorial status whatsoever. They are in and are a part of the host country.
But they have special protections under the Vienna Convention.
I don't know about his other leaks but there's evidence that he communicated with, arguably collaborated with and advised, Manning before Manning copied the classified material. If Manning just sent the material to Wikileaks, Assange would be in the clear (e.g. the Pentagon Papers, Trump's 1995 tax return info this week). Maybe it's not a solid enough case to be sure of a conviction, I doubt the U.S. would want to take a chance on it, but it's not like there's nothing there.
No one has suggested charging him with treason. He certainly can be guilty of espionage against the US without being a citizen.
(And his citizenship is irrelevant to whether he can exceed the protection of the 1st Amendment, since that applies whether or not he is a citizen with regard to government action such as prosecution for espionage.)
Politicians calling for his death is just grandstanding. In fact, the possibility of the death penalty is probably the only thing that would keep Sweden from extraditing him.