Old English flæsc "flesh, meat, muscular parts of animal bodies; body (as opposed to soul)," also "living creatures," also "near kindred" (a sense now obsolete except in phrase flesh and blood), from Proto-Germanic *flaiska-/*fleiski- (source also of Old Frisian flesk, Middle Low German vlees, German Fleisch "flesh," Old Norse flesk "pork, bacon"), which is of uncertain origin; according to Watkins, originally "piece of meat torn off," from PIE *pleik- "to tear," but Boutkan suspects a northern European substratum word.
Of fruits from 1570s. Figurative use for "carnal nature, animal or physical nature of man" (Old English) is from the Bible, especially Paul's use of Greek sarx, and this led to sense of "sensual appetites" (c. 1200).
Flesh-wound is from 1670s; flesh-color, the hue of "Caucasian" skin, is first recorded 1610s, described as a tint composed of "a light pink with a little yellow" [O'Neill, "Dyeing," 1862]. In the flesh "in a bodily form" (1650s) originally was of Jesus (Wyclif has up the flesh, Tindale after the flesh). An Old English poetry-word for "body" was flæsc-hama, literally "flesh-home." A religious tract from 1548 has fleshling "a sensual person." Flesh-company (1520s) was an old term for "sexual intercourse."
The sense of "spurt, rush out suddenly, flow with force" (1540s, usually of water) probably is the same word, with the connecting notion being "sudden movement," but its senses seem more to fit the older ones of flash (v.), now all transferred to this word except in flash flood, via its variant flushe. OED considers this probably not connected to Old French flux. Transitive sense "cause to flow" is from 1590s.
Meaning "cleanse (a drain, etc.) with a rush of water" is from 1789. Of the face, "become suffused with warm color," from 1680s (flushed). Sense of "inflame with pride or passion" as a result of success, victory, etc., is from 1630s; perhaps influenced in sense by flesh (v.). Related: Flushed; flushing.