The bikes will end up dirty, since you'll undoubtedly drive through some weather and slushy roads at this time of year. No biggie.

Taking bikes into hotels is probably fine - I do it all the time. I just walk in with the bike as if it's the most normal thing in the world, and no one ever says anything.

My biggest concern would be when stopping for food, restrooms, etc. Make sure the bikes are locked securely to the rack.