If so, can it be done via the Mini DisplayPort to VGA Adapter I already own?
If TV has VGA input, YES.
Not ALL Air has the same output port however, an not versed on all Air's literations.
I have a 2011 and I believe 2012 also both have thunderbolt ports so u must use a thundebolt adapter. Thunderbolt to VGA or Thunderbolt to hdmi.
The previous Air I believe had a DVI port (a hdmi port without audio), so for those u need a plain DVI cable and a separate audio cable.
Of course u can always go, the heck with cables, and Mirror (wirelessly) from your 11/12 Air to an HDTV via an Apple TV box.
Watch out with some of the cheap adapters. They have trouble transmitting audio. Make sure you read reviews before you buy.