Rigid foamboard is the least likely to become a mouse condo. It is also the most expensive. Styrofoam board or fiberglass/rockwool batts are fine too, and if your construction is tight the mice probably won't get in.
would insulation in the floor draw alot of moisture from litter and spills and cause problems? would foam board sandwitched between two thin peices of plywood work or do mice eat through that stuff and make tunnels.
Personally, in this climate I would not make a raised floor on anything but a tiny coop (in which case, I would heavily insulate the underside with rigid foamboard, I'm not sure whether I'd even bother covering the underside of it with anything, prolly depend what I had kickin' around at the time)
No reason that insulating under the floor should cause dampness or anything btw.
For a normal-sized coop though personally I would put the floor right on the ground -- dirt floor, pavers, concrete, whatever. This plus heavy bedding plus piling snow around the edges of the coop when possible (to insulate the surrounding soil as much or more than to insulate the coop walls) is probably about as good as you're going to get.