Kaissiber 20 has a long article on the Halloween Gambit (p22-51), and mentions briefly that white has even tried 'Die Verfeinerung 4.a3', giving the game Carlsen-Nyysti Finland 2002 as example. (see link below) 

(Yes, it's *the* Carlsen, though in 2002 his rating wasn't all that terrifying yet - still, it's an indication that the whole idea might not be all that dubious...)

http://www.shakki.net/4si/games/gra2.html

Btw, Kaissiber also contains a report from a corr theme tournament, and the theoretical improvements found as a result of it. 

One point of 4.a3 g6 5.Nxe5 is that the knight retreat to g6 was the defence that scored best for black (Ng6 + 7...d6 +2 =4 -12 and Ng6 + 7...d5  +0 =4 -6 (from white's point of view)) and obviously that ain't possible with a3 g6 inserted...

There are lots of names and lots of analysis.

However, it's utterly harmless after 5...Nxe5 6.d4 Nc6 7.d5 Bg7! 8.dxc6 bxc6, which is almost identical to 4...d5 5.exd5 Nxd5 6.Bg2 Nxc3 7.bxc3 (which again is identical to 4...Nxe4 5.Nxe4 d5 6.Nc3 d4 7.Bg2 dxc3 8.bxc3). 

It is hard to imagine how the extra a-pawn move may become significant.