The iPad Mini 2 should work on the AT&T network. You will probably have to take it in to them for a new SIM and provisioning. To the best of my knowledge no carrier (including Verizon) has been locking iPads. But if they did, you have the right to have the device unlocked as long as you own it outright (it is not part of an active carrier contract.)
As for hardware, all recent iPads, including the Mini 2, are capable of working with all the major US carrier networks, and most overseas ones. Only the SIM needs to be changed.
As far a a universal SIM, the only one I know of is Apple's SIM, and I don't think you can buy them separately. They come as an option with the newest iPads when purchased directly from Apple. Even then, some carriers will lock the SIM when you choose one of their plans. (Can't remember if this was AT&T or Verizon).