The diplomatic immunity was a result of Asgard itself becoming an embassy.
Asgard personally saw to it that two mortals (Americans) represented Midgard in the Council of Realms. Not quite an embassy on Asgard, but close. So there’s technically no embassy on Asgard, there’s also no reason to put an embassy in an embassy.
Asgard has entered several treaties with several realms.
Theoretically there is reciprocity as Asgard has had no reason to be angry at someone with diplomatic immunity in the US, but if it came to it they would probably do what they wanted.
I may have misspoken when I said all Asgardians get diplomatic immunity since Stark says “You would get diplomatic immunity” to Thor so he may have been speaking about Thor or Thor and his people.
Yes he would still be guilty of crimes, but he couldn’t be arrested, charged, or prosecuted. So he’s kind of not bound to the law. Sure, they could expel him from the country, but he’s got the whole universe. Not much of a punishment
Diplomatic immunity just guarantees that certain high ranking officials will not be harassed, investigated on, detained or jailed, and they are assured safe passage back to their own country. It's not a blanket exemption from criminal liability.
Embassies on the other hand are deemed extensions of the territories of their counties, so the host country cannot exercise juristiction over the embassy of a certain country because they are just respecting the territorial sovereignty of another. For example, the Embassy of the US in Germany, the German government is powerless to arrest someone inside the embassy of the US because that area is deemed to be an extension of American territory.
If Asgard, while on Earth is given an embassy status, that just means the UN is respecting the territorial sovereignty of Asgard. Not that they are given diplomatic immunity per se. Regarding what Tony Stark said to Thor, I do not know what kind of clout he has that he can assure Thor such privilege that requires the consent of either the host country (which is the US, and any treaty or agreement can only be entered into by the Executive branch of government...aka the President, subject to Legislative approval in certain cases), or by the United Nations (which requires the concurrence of member states).