Nope, you cannot choose wich character you'll get at the shop, except you can choose from subparts of the entire catalogue (Danger = Gheist, Nightmare, Montana.., Cool = ROots, Sakhrom etc...).
You can check out if your characters have been sold using Market, then Successfull sales.
You should know that every 3 days, unsuccessfull publics sales are removed from the market (and the characters goes back to your collection).