During setup, I set my basic and prizes, but my opponet mulligans. I draw my card, and it is another basic. Am I allowed to put it down (face down) and/or switch it the the face down active?
You can put it face down on your bench, but you can't change it with the Pokemon that you put in the Active postion.
That remains the Pokemon that you start with as your Active.
... and just because someone may wonder, if by chance you have placed your Active and 5 Pokemon on your Bench during setup (that would have been quite an opening hand) you cannot switch out one of the Benched Pokemon if you draw a better one for a Mulligan.