Sense of "home for the aged and terminally ill " is from 1879; hospice movement first attested 1978.
"person who receives guests," especially for pay, late 13c., from Old French oste, hoste "guest, host, hostess, landlord" (12c., Modern French hôte), from Latin hospitem (nominative hospes) "guest, stranger, sojourner, visitor (hence also 'foreigner')," also "host; one bound by ties of hospitality."
This appears to be from PIE *ghos-pot-, a compound meaning "guest-master" (compare Old Church Slavonic gospodi "lord, master," literally "lord of strangers"), from the roots *ghos-ti- "stranger, guest, host" and *poti- "powerful; lord." The etymological notion is of someone "with whom one has reciprocal duties of hospitality" [Watkins]. The biological sense of "animal or plant having a parasite" is from 1857.
Spelling evolution influenced by Old Norse cognate gestr (the usual sound changes from the Old English word would have yielded Modern English *yest). Meaning "person entertained for pay" (at an inn, etc.) is from late 13c. Old English also had cuma "stranger, guest," literally "a comer." Phrase be my guest in the sense of "go right ahead" first recorded 1955.
mid-13c., "shelter for the needy," from Old French hospital, ospital "hostel, shelter, lodging" (Modern French hôpital), from Late Latin hospitale "guest-house, inn," noun use of neuter of Latin adjective hospitalis "of a guest or host" (as a noun, "a guest; the duties of hospitality"), from hospes (genitive hospitis) "guest; host;" see host (n.1).
The sense of "charitable institution to house and maintain the needy" in English is from early 15c.; the meaning "institution for sick or wounded people" is recorded by 1540s. The same word, contracted, is hostel and hotel. The sense shift in Latin from duties to buildings might have been via the common term cubiculum hospitalis "guest-chamber." The Latin adjective use continued in Old French, where ospital also could mean "hospitable" and ospitalite could mean "hospital."
Old English open "not closed down, raised up" (of gates, eyelids, etc.), also "exposed, evident, well-known, public," often in a bad sense, "notorious, shameless;" from Proto-Germanic *upana-, literally "put or set up" (source also of Old Norse opinn, Swedish öppen, Danish aaben, Old Saxon opan, Old Frisian epen, Old High German offan, German offen "open"), from PIE root *upo "under," also "up from under," hence also "over." Related to up, and throughout Germanic the word has the appearance of a past participle of *up (v.), but no such verb has been found. The source of words for "open" in many Indo-European languages seems to be an opposite of the word for "closed, shut" (such as Gothic uslukan).
Of physical spaces, "unobstructed, unencumbered," c. 1200; of rooms with unclosed entrances, c. 1300; of wounds, late 14c. Transferred sense of "frank, candid" is attested from early 14c. Of shops, etc., "available for business," it dates from 1824.
Open-door in reference to international trading policies is attested from 1856. Open season is recorded by 1895 of game; figuratively (of persons) by 1914. Open book in the figurative sense of "person easy to understand" is from 1853. Open house "hospitality for all visitors" is first recorded 1824. Open-and-shut "simple, straightforward" first recorded 1841 in New Orleans. Open-faced, of sandwiches, etc., "without an upper layer of bread, etc.," by 1934. Open marriage, one in which the partners sleep with whomever they please, is by 1972. Open road (1817, American English) originally meant a public one; romanticized sense of "traveling as an expression of personal freedom" first recorded 1856, in Whitman.