Short answers, pulled from the story.
A diplomatic mission is fundamentally a group of people from one state present in another state to represent their home country officially. In practice, this phrase usually denotes an embassy or high commission which acts as the main office of a country's diplomatic representatives.
These missions are typically located within the receiving state's capital city though exceptions exist. Some cities host multiple missions from the same country serving different purposes simultaneously such as Rome where many states maintain separate missions to both Italy and the Holy See without sharing premises or personnel.
Diplomatic premises do not enjoy full extraterritorial status as sovereign territory of the represented state. Only a minority of countries grant embassies true sovereign status while most remain under host state jurisdiction.
Consulates differ significantly as smaller diplomatic missions normally found in major cities outside the capital. A consulate-general operates in a major city usually not the capital providing a full range of consular services.
Host country authorities cannot enter mission premises without permission except during emergencies like fires or disasters requiring prompt action. International rules designate any attack on an embassy as an attack on the country it represents.