Old English swete, Mercian swoete, "pleasing to the senses, mind or feelings; having a pleasant disposition," from Proto-Germanic *swotja- (source also of Old Saxon swoti, Old Frisian swet, Swedish söt, Danish sød, Middle Dutch soete, Dutch zoet, Old High German swuozi, German süß).
This is reconstructed to be from the PIE root *swād- "sweet, pleasant" (source also of Sanskrit svadus "sweet;" Greek hedys "sweet, pleasant, agreeable," hedone "pleasure;" Latin suavis "pleasant" (not especially of taste), suadere "to advise," properly "to make something pleasant to").
As "pleasing to the ear" in Old English; in reference to jazz played at a steady tempo and without improvisation, by 1924 (opposed to hot). As "pleasing to the eye, beautiful, desirable" mid-14c. Of persons, "gracious, kind, having pleasant manners," in Old English. Words for "sweet" in the Indo-European languages typically are used in reference to the other sense as well, and in general for "pleasing."
Also "being in a sound or wholesome state" (mid-13c.), and, of water, "fresh, not salt" (late Old English). In old chemistry, "free from corrosive salts, acids, etc.," 1660s. Hence in the oil industry, in reference to petroleum, "free from sulfur compounds" (1919).
As "dear to oneself," sometimes sarcastic, by 1620s (take my sweet time). As an intensifier in coarse slang (sweet f-all), by 1958. Sweet Jesus as an imprecation is by 1932.
To be sweet on (someone) "infatuated with" is by 1690s. Sweet in bed (c. 1300) "has been used with various implications" [OED, 2nd ed. print 1989]. Sweet sixteen is recorded by 1767, in reference to the age, usually of girls.
Then come kiss me, sweet-and-twenty!
Youth's a stuff will not endure.
["Twelfth Night"]
A phrase, not a title. Sweet dreams as a parting to one going to bed is attested from 1897, short for sweet dreams to you, etc. Sweet-and-sour in cookery is from 1723 and not originally of Eastern dishes. Sweet nothings "sentimental trivialities," generally whispered in another's ear, is from 1900.
Sweet spot is from 1976, in reference to tennis rackets and golf swings. Sweet thing "desirable object or person" is by 1887. Sweet corn "sweetened maize" is from 1640s. Sweet singer "religious poet" is by 1550s, especially sweet singer of Israel, adapted from 2 Samuel xxiii.1.
The word in Chaucer's shoures soote (or schowres swoote) is Middle English sote "sweet to the senses," from Old English swot, a different form from the same Germanic adjective.