Mice generally stick to shelters. Getting away from one is the key to avoiding mice. Ants? Much more of a problem with a tarp or tent than a hammock. Truthfully, possums and skunks and occasional bugs are the only issues I've ever had with critters, even on the ground. No problems at all in a hammock.

On a separate note, if you're looking at a long-distance hike (and with the mention of shelters, I'm guessing AT), get yourself a much larger tarp than the diamonds that come with a Hennessy or with ENO hammocks. With an 8' x 10' tarp, (which it sounds like you may already own) you'll have much better protection, more privacy (to change clothes) and a generally better experience.

My one negative experience with a hammock was being in 5 days of rain with a stock fly. Cooking under it is a pain since you keep brushing into the hammock and worrying about an errant flame melting a bit of your bed. For tarps, bigger is generally better for hammocking.