mid-13c., flusshen "move rapidly or violently; rush, dart, spring" (intransitive); late 15c., flush up, transitive, "cause to fly; start or flush (birds)," perhaps imitative of the sound of beating wings.
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.
1550s, "perfect, faultless;" c. 1600, "abundantly full," also "full of life or spirit," also "plentifully supplied" (with money, etc.), perhaps from flush (v.1) through the notion of a river running full, hence level with its banks. Meaning "even, level" is from 1620s, originally of ship's decks. In general use by 1791; in typography, 1900; in pugilism, 1812.
The section of entries for the various flushes in Century Dictionary opens with a caveat:
The several words spelled flush, being mostly dialectal, colloquial, or technical, and scantily recorded in early literature, have become partly confused with one another, and cannot now be entirely disentangled. Words originally different have acquired some meanings very nearly identical, while on the other hand there are some meanings not obviously related which are, nevertheless, to be referred to one original.
Weekley calls it "A very puzzling word." Sense of "a rush of water" in a stream (1520s), is probably from flush (v.1). From this likely come the extended senses "rush of emotion or passion" (1610s); "a sudden shooting up" (1773); "act of cleansing (a drain) by flushing" (1883); "glow of light or color" (especially sudden redness in the face), 1620s. Independently from the verb, probably, is the noun sense of "a flight of birds suddenly started up" (1590s).
The meaning "hand of cards all of one suit" (1520s) is of uncertain origin, perhaps formed on the model of French flus (15c.), from Old French flux, flus "a flowing, rolling" (see flux), which, in common with its Italian cognate flusso, is said to have once had a sense of "a run" of cards. The form in English probably was influenced by flush (v.1).