List of Dosbox games:
http://www.dosbox.com/wiki/GOG_games_that_use_DOSBox You can play them windowed (press Alt+Enter ingame), which seems very helpful in your situation. Personally I wouldn't go for an action game, an adventure, rpg or strategy game may be easier to switch and don't need the sound turned on. So, take a look on Mater of Orion pack or the first Gabriel Knight. Also todays release Lands or Lore looks interesting, it's also use Dosbox and the first is a very newbie-friendly rpg games. I only choosed the cheap one for 6 bucks.
Edit: ScummVM games on gog: Beneath a steel sky (free, highly recommended), the first two Broken Sword games (also highly recommended), Feeble Files, Gobliins pack, Lure of the Temptress (free), and the first two Simon the Sorcerer games. Maybe I miss some.