What a cool looking place, inside and out. Big congrats!!
Judging from the pics I see nothing a few bucks and elbow grease won't take care of. As Bill mentioned I'd wrap the fascia in aluminum. Check with the local lumber yard or ReStore outfit if you have one near by. They might have some leftovers you can pickup on the cheap. Easy to work with and a one and done fix. Unless you want to clean gutters twice a year I wouldn't waste my time with them, your call.
I wouldn't worry too much about putting up lattice around the bottom, if that's what your talking about. It won't do much for keeping out mice. My cabin is constructed the same way and I have not had any issues with larger vermin under it...so far.
Battling mice is a constant. I use the block poison inside and out. I use plastic containers with a screw on lid. Cut a small hole in them for access and then bait it. Keeps the mess down to a minimum. I leave the lid on the toilet open so mice head for the water and die in the rv antifreeze. I've seen as many as 6 dead at one time. The bucket trick works great too. Taking out a few trees and keeping the grass mowed short around the place would be a big help against mice also.
Good luck!!