no, the Samsung galaxy axiom is a CDMA only phone, T-Mobile requires a GSM phone
yes, the Galaxy S3 Mini is a GSM phone, and will work, but will likely be limited to Edge (2g) speeds.
You cannot unlock your samsung galaxy axiom because your samsung galaxy axiom is a cdma network. But you can easily unlock your samsung galaxy 3 mini. Because it is a gsm phone.
If you need to unlock your samsung galaxy s3 mini first can visit this site Prounlocking.com here you can get the code and unlock your samsung galaxy 3 mini easily.