Here's an article that goes into more detail on what international law says about embassies [1].
I'm not sure if the NYT enforces their paywall on that article--it works for me and I'm not a subscriber, but I am signed up for one of their email lists and it is possible that's why I can see it.
For those who get stopped by the paywall TL;DR would be that the international law for embassies and other diplomatic facilities is all about what the "receiving state" can and cannot do. The "receiving state" is the state that the embassy is in.
As far as other states go there is nothing legally special about embassies. So if country X attacks country Y's embassy that is located in country Z as far as international law goes it is pretty much the same as attacking any other random building in Z that happens to have people and stuff from country Y inside.
That is not correct. Protection of embassies only applies against the host country.