What is the name for a mouth guard with air pockets?

Depands. Your question is a bit unclear. Your orthodontist will typically, when you are done with braces, use a plastic retainer that we call an "essix retainer". We also use mouthguards like the one pictured by the other doc that has space for braces.
Air Guard. Made by safe-t-guard, I have seen it, but haven't seen any studies that show it to be more or less effective than a real custom made mouth-guard by a dental office. They are made on a special type of vac-u-forming machine in layers for more protection.
Not sure. I know of one that is filled with fluid called an aqualizer...
Bleaching trays? Not 100% sure what you are referring too, but perhaps you mean bleaching trays. They are similar to mouth guards but have spaces or "air pockets" in the area of the front teeth to accommodate the bleaching gel used for tooth whitening.