I'm not sure if mice would be interested in the worm bin as the bedding should be kept moist and that might deter them from wanting to snuggle in for the winter. As to if they would go forraging for food, I'm not sure. I don't know what kind of worm bin you are getting but a bit of hardware cloth could probably be arranged to protect your worm bin from mice if it turns out to be a problem.

I don't think the mites in a worm bin are like what bother animals but I don't know if the mites that bother animals might decide to take up residence in a worm bin if one happens to be nearby. I do know that many people who raise rabbits do set up worm bins under the cages to let the worms process the bunnie berries. If this tended to increase mite problems in the rabbits I don't think it would be quite so common to hear about. I doubt the worm bin you ordered will be the right shape to utilize under the rabbit cages but you might decide you want to try something like that in the future if you like your worm bin you have on the way.