air (v.) Look up air at Dictionary.com
"to expose to open air," 1520s, from air (n.1). Figurative sense of "to expose, make public" is from 1610s of objects, 1862 of opinions, grievances, etc. Meaning "to broadcast" (originally on radio) is from 1933. Related: Aired; airing.
air (n.1) Look up air at Dictionary.com
c.1300, "invisible gases that make up the atmosphere," from Old French air "atmosphere, breeze, weather" (12c.), from Latin aerem (nominative aer) "air, lower atmosphere, sky," from Greek aer (genitive aeros) "air" (related to aenai "to blow, breathe"), of unknown origin, possibly from a base *awer- and thus related to aeirein "to raise" and arteria "windpipe, artery" (see aorta) on notion of "lifting, that which rises." In Homer mostly "thick air, mist;" later "air" as one of the four elements.

Words for "air" in Indo-European languages tend to be associated with wind, brightness, sky. In English, air Replaced native lyft, luft (see loft (n.)). To be in the air "in general awareness" is from 1875; up in the air "uncertain, doubtful" is from 1752. To build castles in the air is from 1590s (in 17c. English had airmonger "one preoccupied with visionary projects"). Broadcasting sense (as in on the air) first recorded 1927. To give (someone) the air "dismiss" is from 1900. Air pollution is attested by 1870.
air (n.2) Look up air at Dictionary.com
1590s, "manner, appearance" (as in an air of mystery); 1650s, "assumed manner, affected appearance" (especially in phrase put on airs, 1781), from French air "look, appearance, mien, bearing, tone" (Old French aire "reality, essence, nature, descent, extraction," 12c.; compare debonair), from Latin ager "place, field" (see acre) on notion of "place of origin."

But some French sources connect this Old French word with the source of air (n.1), and it also is possible these senses in English developed from or were influenced by air (n.1); compare sense development of atmosphere and Latin spiritus "breath, breeze," also "high spirit, pride," and the extended senses of anima.
air (n.3) Look up air at Dictionary.com
"melody, tune," 1580s, from Italian aria (see aria).