gomba, this is the drag racing section. Different tires on the front is pretty much a given.
Running different sized tires front to back is fine on the street, even different sizes and different brands. Yes, behavior at the limit on the street with change, but that's to be expected and hopefully part of the reason you're doing it in the first place. If both types of tires have similar rain performance, you'll not see any drastic changes there either. If they have largely different wet traction, then just make sure the rears have more wet traction that the front to give you that "safe" understeer.