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 aer "air, lower atmosphere, sky," from Greek aer (genitive aeros) "air" (related to aenai "to blow, breathe"), which is of unknown origin.

It is possibly from a PIE *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.)).

In old chemistry, air (with a qualifying adjective) was used of any gas. 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, airplay) first recorded 1927. To give (someone) the air "dismiss" is from 1900. Air pollution is attested by 1870. Air guitar by 1983. Air traffic controller is from 1956.
air (v.) Look up air at Dictionary.com
1520s, "expose to open air," 1520s, from air (n.1). Figurative sense of "expose ostentatiously, 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.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), which is perhaps from Latin ager "place, field, productive land" (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, connected rhythmic succession of distinct musical sounds," 1580s, nativized from Italian aria (see aria), perhaps via French.