Short answers, pulled from the story.
A diplomatic mission is the broader term for any official group of representatives from one state present in another. An embassy is one specific type of diplomatic mission, usually located in the capital of the receiving state and providing a full range of services including consular work. The term also covers high commissions, legations, consulates, and permanent missions to international organizations.
No. Contrary to popular belief, embassy premises remain under the jurisdiction of the host state and are generally not sovereign territory of the represented country. The sending state can grant embassies sovereign status, but only a minority of countries do so. The Vienna Convention on Diplomatic Relations provides immunity from most local laws and bars host authorities from entering without permission, but the land itself stays under host-state jurisdiction.
The embassy is the diplomatic delegation itself, meaning the people who make up the mission. The chancery is the physical office space where diplomatic work is carried out. Technically, the embassy operates in the chancery, though the two terms are often used interchangeably in everyday speech.
A chargé d'affaires is the official who heads a diplomatic mission when the chief of mission has been recalled or is absent. Recalling a head of mission is a common diplomatic signal of displeasure that falls short of cutting relations entirely. A chargé d'affaires ad interim also leads the mission during the gap between one chief of mission's departure and a successor's appointment.
Three cases are widely cited. The British Embassy in Beijing was invaded repeatedly in 1967. The American Embassy in Tehran was the site of a hostage crisis that ran from 1979 to 1981. The Japanese ambassador's residence in Lima, Peru, was seized in a hostage crisis that lasted from 1996 to 1997.
Unrecognized governments and non-sovereign territories cannot establish formal diplomatic missions under the Vienna Convention. Instead, they set up non-diplomatic offices that promote trade and assist their citizens without carrying official status. Examples include the Taipei Economic and Cultural Representative Offices for the Republic of China, and Somaliland's representative offices in London, Addis Ababa, Rome, Taipei, and Washington, D.C.