Leave the skunk set!! Good medicine for a coyote. Should make it more appealing to a coyote. I never pull a skunked set.

As for the badger. Find his den. They are easy to catch with a blind set at the mouth of the den. You should be able to see where he is walking and just bed a trap at the mouth of the den. I would use a #3, 4 coiled trap for the badger. Put your stake directly under the trap!! Badgers will dig furiously when caught. They pull to the end of the chain and dig as deep as they can all the way around the circle. You will be shocked when you see the mess if you catch him.