Just bought a pair of Nike MTB shoes.. The MTB pedals I bought ask that you install the insole washer and nut (supplied with the pedals). However, my shoes seem to already have an existing insole washer/nut. Which would should I use?

Use the ones that come with the shoes but the screws may be too short. I also find many times the screws they give you with the pedals are too long and poke through the insole plate and stick you in the bottom of the foot for longer rides. For those times I either pick up shorter screws at the LBS or I cut them shorter with a fine tooth hacksaw.