1510s, "to send off, send to a destination," usually implying urgent importance or haste, from Spanish despachar "expedite, hasten" or cognate Italian dispacciare "to dispatch." For first element, see dis-.
The second element apparently has been confused or corrupted, and its exact source and meaning is uncertain. One proposal is that it is Vulgar Latin *pactare "to fasten, fix" or *pactiare. Another says it is Latin -pedicare "to entrap" (from Latin pedica "shackle;" see impeach), and the Spanish and Italian words seem to be related to (perhaps opposites of) Old Provençal empachar "impede." See OED for full discussion.
Meaning "get rid of promptly by killing" is attested from 1520s; that of "attend to, finish, bring to an end, accomplish" is from 1530s. Related: Dispatched; dispatching.
1540s, "dismissal after settlement of business," from dispatch (v.). Meaning "speed, haste" is from 1570s. Sense of "a written message sent speedily" is first attested 1580s; that of "a sending off or away" is from c. 1600.
"a blow," c. 1300, of uncertain origin, older than the verb, possibly related to Middle Dutch boke, Middle High German buc, and Danish bask, all meaning "a blow;" perhaps imitative; perhaps from some sense of box (n.1) or (v.2).
"rectangular wooden container," usually with a lid, Old English box, also the name of a type of shrub, from Late Latin buxis, from Greek pyxis "boxwood," pyxion "writing table, box," made of boxwood, from pyxos "box tree," which is of uncertain origin. Beekes suggests a loan-word from Italy, as that is where the tree is native. Dutch bus, German Büchse "box; barrel of a gun," also are Latin loan-words.
Meaning "compartment at a theater" is from c. 1600 (box seat in the theatrical sense is by 1850). Meaning "pigeon-hole at a post office" is from 1832. Meaning "television" is from 1950 (earlier "gramophone player," 1924). Meaning "station of a player in baseball" is from 1881. Graphics sense "space enclosed within borders and rules" is from 1929. Slang meaning "vulva" is attested 17c., according to "Dictionary of American Slang;" modern use seems to date from c. World War II, perhaps originally Australian, on notion of "box of tricks." Box lunch (n.) attested from 1899. The box set "multiple-album, CD or cassette issue of the work of an artist" is attested by 1955. To think or act outside the box "contrary to convention" is attested by 1994.