Embassies are still part of the host country, but generally (a) police/military of the host country cannot enter without permission of the Ambassador and (b) most employees who came from the mission country hold a diplomatic passport or have (some) diplomatic immunity by treaty, making them immune to prosecution, and (c) some laws of the host country apply to the embassy employees or operate on the embassy functions, but this would be moreso about procedure and secrecy than mundane daily life laws.
There may be exceptions to the above in treaties. In the UK there are exceptions to getting permission to entering an embassy (legal exceptions -- obviously the political fallout would be up for debate).
If two people making a visa application got into a fight in line, resulting in major injury, the Ambassador would probably just call the local police and have them dealt with locally, because the host country's laws still apply. However if one of the guards shot a local troublemaker who appeared to be a security threat, the Ambassador might simply do nothing, resulting in the guard not being able to be arrested.
https://en.wikipedia.org/wiki/Diplomatic_mission#Extraterrit...
https://en.wikipedia.org/wiki/Diplomatic_immunity
US Marine Security Guards carry a certain level of diplomatic immunity https://en.wikipedia.org/wiki/Marine_Security_Guard#Responsi...