where you live, there is probably lots of moisture both winter and summer, so no matter if you use felt, foam, or anything else, you're going to have to air it out and/or change it regularly.
for sound insulation, i'd get something like the Dynamat stuff they sell on ebay. i guess you can buy similar stuff lots cheaper in Menards, Home Depot, etc, as it's the same thing they use for roof repairs: foil covered asphalt/adhesive on wide rolls.
the other thing i'd use is a large piece of a carpet remnant. cut the pieces to fit both sides of your floor and put them on top of the rubber mat. then all you have to do is pull them out and throw them away when they get too soiled. just cut new ones as you need them. you'll still need to pull put the rubber mat from time-to-time to let everything under it dry real good and prevent rust.
that's the way i do it here as we got so much sand/salt in the winter on the snowy roads.