The UT course differs from the NRA course and must be given by a UT certified instructor. Not sure about the fingerprints, but at the very least, I would assume that you would have to obtain them from the authorities in FL and forward them to the authorities in UT. (probably not worth the hassle.)
Not sure about long gun purchases in MN, however in-person non-resident handgun purchases are against federal law. If you wish to purchase a handgun from a state in which you do not reside, it must be shipped to a FFL in your home state, at which point a BG check must be performed as though you were purchasing it from that FFL.