c. 1200, sete, "thing to sit on; place one sits," from Old Norse sæti "seat, position," both from Proto-Germanic *sæt- (source also of Old High German saze, Middle Dutch gesaete "seat," Old High German gisazi, German Gesäß "buttocks"), from PIE root *sed- (1) "to sit." Old English had sæt "place where one sits in ambush," which also meant "residents, inhabitants," and is the source of the -set in Dorset and Somerset.
The sense of "part of a thing (a saddle, etc.) on which one sits" is from c. 1400. The meaning "posterior of the body" (the sitting part) is from c. 1600; the sense of "part of a garment which covers the buttocks" is from 1835. Seat belt "safety restraint when sitting" is from 1915, originally in airplanes.
By late 14c. as "part of the body in which a humor arises;" from 1550s as "site, situation, location" generally.
The word in the sense of "residence, abode, established place" (late 13c.) is an extended use of this, influenced by Old French siege "seat, established place," and Latin sedes "seat." It is perhaps from the notion of a chair set apart for the holder of some position of dignity or authority (a sense attested in English seat from c. 1200). The meaning "city in which a government sits" is attested from c. 1400. The sense of "right of taking a place in a parliament or other legislative body" is attested from 1774.
1570s, of a house, town, etc., "to be in a certain position" (implied in seated), from seat (n.). Of diseases, in the body, from 1610s (hence deep-seated). Transitive sense of "locate, settle, place permanently" is from 1580s.
The meaning "cause to sit, place on a seat" is from 1590s, especially "cause to sit on a throne or other seat of dignity." From c. 1600 as "set or secure in its proper place," hence many extended senses in mechanics. Of a theater, etc., "afford seating accommodations for," by 1830.