The Pokebank feature is completely dependent on your player ID & Nintendo 3DS system (if it were to get stolen, you can sign into your account on a different system after calling customer support. They will transfer things for you, like digital downloads, but not the game cartridge you may have gotten stolen as well). So you can reset your game and keep all your Pokémon. :D
No, they would not act like traded Pokémon. They have your Trainer ID which stays the same even if you reset your game.
Hope I helped!!