"Of somewhat doubtful etymology" [OED]. Liberman calls it "one of many imitative verbs beginning with fl- and denoting unsteady or light, repeated movement" (for example flicker, flutter). If it is related to flat the notion could be either "caress with the flat of the hand, stroke, pet," or "throw oneself flat on the ground" (in fawning adoration). The -er ending is unusual for an English verb from French; perhaps it is by influence of shimmer, flicker, etc., or from flattery.
Meaning "give a pleasing but false impression to" is from late 14c. Sense of "show (something) to best advantage" is from 1580s, originally of portraits. Related: Flattered; flattering.
"soothing, allaying irritation;" as a noun, "a medicine which assuages the effects of irritation," 1732, from Latin demulcentem (nominative demulcens), present participle of demulcere "to stroke down, soothingly pet," from de "down" (see de-) + mulcere "to stroke, caress," from PIE *m(o)lk-eie- "to touch repeatedly," source also of Sanskrit mrsase "to touch." De Vaan writes that connection with *meig-, the root of mulgere "to milk," "is possible, but unproven." The obsolete verb demulce "soothe, soften, mollify" is attested from 1520s.