No.
Yes, but the overwhelming majority of security is handled by contractors. Usually locals in secure countries, foreign contractors in unsecure countries.
Every embassy has a Marine Security Detachment. Some may be very small (less than 10 Marines) and others might be very large (100+). The Marine Security Detachment is responsible for internal security - everything from directing visitors to wear their Visitor badge to checking door locks and file cabinets to see that stuff is locked up tight at night. They also form the Honor Guard when the Ambassador receives foreign dignitaries. However, their primary responsibility is to protect American soil from invasion - security inside the embassy perimeter. They stand inside the gates and tell foreign nationals they cannot come inside for sanctuary against roving bands of foreign nationals (representing the country where the embassy is located or rebel/outlaw bands) go about massacring folks. (OK, that's extreme, but it has happened.)
Security outside the embassy can be provided by the forieign government (military, civillian or contract staff). Those are, for example, the Lybians who drifted off and away from in front of the consulate at Benghazi. In addition, the Ambassador/State Dept. may contract for private security forces to operate outside the embassy perimeter (and occassionally inside the perimeter as well).
On the flip side, the State Dept. Protective Service, uniformed division, is tasked with providing security in front of all embassies in DC. That includes EPS officers stationed in front of the empty Iranian structure whose ownership is still unsettled as to the Shah or the bunch of rebels who threw him out. IIRC they have concurrent jurisdiction with DC Metro from the center of the roadway in front of the foreign embassy to the perimeter inside the sidewalk. Consulates are generally "protected" by the local police dept.
Crimes taking place inside an embassy are the jurisdiction of the FBI - which is why they end up in foreign countries. The FBI has no jurisdiction for crimes occurring in front of foreign embassies/consulates, but does have jurisdiction in crimes committed against federal employees.
When I was a motorcycle courier in DC we used to play a game of quizzing each other about what agency would have jurisdiction over our moving violations - mostly to see if we could arrange for the wrong agency to cite us.
stay safe.
