For the Challenge Medals can I not use Dual-Type Pokemon. Say for the Fire-type Champ: could I not use a Blaziken because it is also Fighting or does it have to only be Fire?
It is perfectly okay to use dual typed Pokemon. I myself have used, say, Cradily for my Grass run and then again for Rock. I also did my One and Only challenge with Gyarados and received the Water and Flying medals too.
So, you could use a Blaziken for both the Fire and Fighting medals. All that matters is that every Pokemon you use has at least one type in common.