This idea sounds promising but needs some more work to get me excited.
The problem with carpets it that it's generally not a good idea to have one if you're allergic. Regular carpets get saturated with dust as time goes by. Regular vacuum cleaners are far from perfect, they leave an increasing amount of dust behind, yet every time you walk over the carpet, a tiny amount of dust is released. From my own experience, it works ok to get rid of carpets and simply mop up the place on a regular basis.
So basically the problem that has to be solved is that the carpet needs to release the dust/hair on demand. Maybe a tiny electric current would work that makes the carpet electrostatic - then simply turn it "off" for vacuum cleaning.
Either way the carpet would need to be rather flat (not fluffy) and made of a special material that doesn't provide a home for mites.