Pretty much anything that dissolves in lipids will work as an anesthetic, with effectiveness depending on how well it dissolves. This is why nitrogen under high pressure can cause nitrogen narcosis while diving. This is also why spot removers like ether and chloroform work as anesthetics. Also why nitrous oxide is used for whipped cream; it dissolves in the cream and makes it foam up. Since xenon is somewhat soluble in lipids, it somewhat works as an anesthetic even though it is chemically inert. The big question is why being lipid-soluble makes something an anesthetic. That's not known, but probably it messes up ion channels in the lipid membrane or something.

See the Meyer-Overton correlation: https://en.wikipedia.org/wiki/Theories_of_general_anaestheti...

It's not completely chemically inert, but the actual mechanism that makes it work as an anesthetic is unknown (as, unnervingly, are most general anesthetics).

It's not chemically inert (1), but even if it were it could still disrupt cell membrane function by physical effects, like dissolving into the lipids.

(1) For example, a mixture of xenon and fluorine, when exposed to sunlight, makes crystals of xenon difluoride.

Nitrous Oxide is inert in the body as well.