fly (n.)
Old English fleoge "fly, winged insect," from Proto-Germanic *fleugjon (cognates: Old Saxon fleiga, Old Norse fluga, Middle Dutch vlieghe, Dutch vlieg, Old High German flioga, German Fliege "fly); literally "the flying (insect)" (compare Old English fleogende "flying"), from same source as fly (v.1).

Originally any winged insect (hence butterfly, etc.); long used by farmers and gardeners for any insect parasite. The Old English plural in -n (as in oxen) gradually normalized 13c.-15c. to -s. Fly on the wall "unseen observer" first recorded 1881. An Old English word for "curtain" was fleonet "fly-net." Fly-swatter as a bit of wire mesh on a handle first attested 1917. Fly-fishing is from 1650s.
fly (v.1)
"to soar through air," Old English fleogan "to fly" (class II strong verb; past tense fleag, past participle flogen), from Proto-Germanic *fleugan "to fly" (cognates: Old Saxon and Old High German fliogan, Old Norse flügja, Old Frisian fliaga, Middle Dutch vlieghen, Dutch vliegen, German fliegen), from PIE *pleu- "flowing, floating" (see pluvial).

Notion of "flapping as a wing does" led to noun sense of "tent flap" (1810), which yielded (1844) "covering for buttons that close up a garment." The noun sense of "a flight, flying" is from mid-15c. Baseball fly ball attested by 1866. Slang phrase fly off the handle "lose one's cool" dates from 1825. To do something on the fly is 1856, apparently from baseball.
fly (v.2)
"run away," Old English fleon (see flee). Fleogan and fleon were often confused in Old English, too. Modern English distinguishes in preterite: flew/fled.
fly (adj.)
slang, "clever, alert, wide awake," late 18c., perhaps from fly (n.) on the notion of the insect being hard to catch. Other theories, however, trace it to fledge or flash. Slang use in 1990s might be a revival or a reinvention.