Old English æt, from Proto-Germanic *at (source also of Old Norse, Gothic at, Old Frisian et, Old High German az), from PIE root *ad- "to, near, at." Lost in German and Dutch, which use their equivalent of to; in Scandinavian, however, to has been lost and at fills its place.
At is used to denote relations of so many kinds, and some of these so remote from its primary local sense, that a classification of its uses is very difficult. [OED]
In choosing between at church, in church, etc. at is properly distinguished from in or on by involving some practical connection; a worshipper is at church; a tourist is in the church. In 19c. it was used for points of the compass as regions of a country (at the South) where later tendency was to use in.
The colloquial use of at after where (as in where it's at) is noted in Bartlett (1859). At last is recorded from late 13c.; adverbial phrase at least was in use by 1775. At in Middle English was used freely with prepositions (as in at after, which is in Shakespeare), but this has faded with the exception of at about.
"perform, execute, achieve, carry out, bring to pass by procedure of any kind," etc., etc., Middle English do, first person singular of Old English don "make, act, perform, cause; to put, to place," from West Germanic *doanan (source also of Old Saxon duan, Old Frisian dwa, Dutch doen, Old High German tuon, German tun), from PIE root *dhe- "to set, put, place."
Use as an auxiliary began in Middle English. Sense of "to put, place, lay" is obsolete except in phrases such as do away with. Periphrastic form in negative sentences (They did not think) replaced the Old English negative particles (Hie ne wendon).
Meaning "visit as a tourist" is from 1817. In old slang it meant "to hoax, cheat, swindle" (1640s). Slang meaning "to do the sex act with or to" is from 1913.
Slang do in "bring disaster upon, kill" is by 1905. To have to do with "have concern or connection with" is from late 13c. To do without "dispense with" is from 1713. Expression do or die indicating determination to succeed despite dangers or obstacles is attested from 1620s.
1550s, "large kind of four-wheeled, covered carriage," from French coche (16c.), from German kotsche, from Hungarian kocsi (szekér) "(carriage) of Kocs," village where it was first made. In Hungary, the thing and the name for it date from 15c., and forms are found since 16c. in most European languages (Spanish and Portuguese coche, Italian cocchino, Dutch koets). Vehicles often were named for the place of their invention or first use (compare berlin, landau, surrey). Applied to railway passenger cars by 1866, American English. Sense of "economy or tourist class" is from 1949.
Meaning "instructor/trainer" is c. 1830 Oxford University slang for a private tutor who "carries" a student through an exam (compare pony in the student slang sense "translation"). Transferred sense in sports, "person employed to train athletes for a contest" is attested from 1861. A more classical word for an athletic trainer was agonistarch, from Greek agonistarkhes "one who trains (someone) to compete in the public games and contests."
All panelled carriages with seats for four persons inside, and an elevated coachman's seat, are designated coaches. The town coach proper, has windows in the doors, and one in each end, the quarters being panelled. [Henry William Herbert ("Frank Forester"), "Hints to Horse-Keepers," New York, 1859]
A sound found chiefly in words of Old English, Old Norse or Greek origin, unpronounceable by Normans and many other Europeans. In Greek, the sound corresponds etymologically to Sanskrit -dh- and English -d-; and it was represented graphically by -TH- and at first pronounced as a true aspirate (as still in English outhouse, shithead, etc.).
But by 2c. B.C.E. the Greek letter theta was in universal use and had the modern "-th-" sound. Latin had neither the letter nor the sound, however, and the Romans represented Greek theta by -TH-, which they generally pronounced, at least in Late Latin, as simple "-t-" (passed down to Romanic languages, as in Spanish termal "thermal," teoria "theory," teatro "theater").
In Germanic languages it represents PIE *-t- and was common at the start of words or after stressed vowels. To represent it, Old English and Old Norse used the characters ð "eth" (a modified form of -d-) and þ "thorn," which originally was a rune. Old English, unlike Old Norse, seems never to have standardized which of the two versions of the sound ("hard" and "soft") was represented by which of the two letters.
The digraph -th- sometimes appears in early Old English, on the Roman model, and it returned in Middle English with the French scribes, driving out eth by c. 1250, but thorn persisted, especially in demonstratives (þat, þe, þis, etc.), even as other words were being spelled with -th-. The advent of printing dealt its death-blow, however, as types were imported from continental founders, who had no thorn. For a time y was used in its place (especially in Scotland), because it had a similar shape, hence ye for the in historical tourist trap Ye Olde _______ Shoppe (it never was pronounced "ye," only spelled that way).
The awareness that some Latin words in t- were from Greek th- encouraged over-correction in English and created unetymological forms such as Thames and author, while some words borrowed from Romanic languages preserve, on the Roman model, the Greek -th- spelling but the simple Latin "t" pronunciation (as in Thomas and thyme).