air:  Modern English air is a blend of three strands of meaning from, ultimately, two completely separate sources. In the sense of the gas we breathe it goes back via Old French air and Latin āēr to Greek áēr ‘air’ (whence the aero-compounds of English; see AEROPLANE). Related words in Greek were áērni ‘I blow’ and aúrā ‘breeze’ (from which English acquired aura in the 18th century), and cognates in other Indo-European languages include Latin ventus ‘wind’, English wind, and nirvana ‘extinction of existence’, which in Sanskrit meant literally ‘blown out’.
In the 16th century a completely new set of meanings of air arrived in English: ‘appearance’ or ‘demeanour’. The first known instance comes in Shakespeare’s 1 Henry IV, IV, i: ‘The quality and air of our attempt brooks no division’ (1596). This air was borrowed from French, where it probably represents an earlier, Old French, aire ‘nature, quality’, whose original literal meaning ‘place of origin’ (reflected in another derivative, eyrie) takes it back to Latin ager ‘place, field’, source of English agriculture and related to acre. (The final syllable of English debonair  came from Old French aire, incidentally; the phrase de bon aire meant ‘of good disposition’.) The final strand in modern English air comes via the Italian descendant of Latin āēr, aria.
This had absorbed the ‘nature, quality’ meanings of Old French aire, and developed them further to ‘melody’ (perhaps on the model of German weise, which means both ‘way, manner’ and ‘tune’ – its English cognate wise, as in ‘in no wise’, meant ‘song’ from the 11th to the 13th centuries). It seems likely that English air in the sense ‘tune’ is a direct translation of the Italian.
Here again, Shakespeare got in with it first – in A Midsummer Night’s Dream, I, i: ‘Your tongue’s sweet air more tunable than lark to shepherd’s ear’ (1590). (Aria itself became an English word in the 18th century.) => acre, aeroplane, agriculture, aria, aura, eyrie, malaria, wind
"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.
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"), which is 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.
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.