Meaning "to throw, hurl, let fly" is from mid-14c. Sense of "set (a boat) afloat" first recorded c. 1400, from notion of throwing it out on the water; generalized by 1600 to any sort of beginning. Related: Launched; launching.
"artificial satellite," extended from the name of the one launched by the Soviet Union Oct. 4, 1957, from Russian sputnik "satellite," literally "traveling companion" (in this use short for sputnik zemlyi, "traveling companion of the Earth") from Old Church Slavonic supotiniku, from Russian so-, s- "with, together" + put' "path, way," from Old Church Slavonic poti, from PIE *pent- "to tread, go" (see find (v.)) + agent suffix -nik.
The electrifying impact of the launch on the West can be gauged by the number of new formations in -nik around this time (the suffix had been present in a Yiddish context for at least a decade before); Laika, the stray dog launched aboard Sputnik 2 (Nov. 2, 1957), which was dubbed muttnik in the Detroit Free Press, etc. The rival U.S. satellite which failed to reach orbit in 1957 (because the Vanguard rocket blew up on the launch pad) derided as a kaputnik (in the Daytona Beach Morning Journal), a dudnik (Christian Science Monitor), a flopnik (Youngstown Vindicator, New York Times), a pffftnik (National Review), and a stayputnik (Vancouver Sun).
"short, erect tail" (of a rabbit, hare, deer, etc.), 1520s; earlier "a hare" (mid-15c., perhaps c. 1300), a word of obscure origin.
Perhaps it is from Old Norse skjota "to shoot (with a weapon), launch, push, shove quickly" (compare Norwegian skudda "to shove, push"), from PIE root *skeud- "to shoot, chase, throw." Or perhaps it is a relative of Middle English sheten "hasten from one place to another," from Old English sceotan, sceotian, from Proto-Germanic *skeutanan (source also of Old Frisian skiata "to shoot, supply," Old Dutch scietan), for which Boutkan offers no IE etymology.
Also compare Middle English scut (v.) "make short, hurried runs," as a noun, "a short garment" (mid-15c.), as an adjective, "short" (c. 1200), perhaps from Old French escorter, from Latin excurtare.