I know that head tilt causes rodents to walk in circles, which is an inner ear infection. As for things you can get from this mouse, I'd be worried about lice, fleas and ticks. When an animal dies (i've witnessed this, its weird) the lice and ticks and mites and all of that have this mass exodus to find a new, warm body to inhabit, because the dead body is no longer appealing. That means you, your cat, your kid... could end up with a bunch of bugs. Some of those bugs can be carrying diseases.
I'd seriously avoid getting bit by it, a wild animal carries more bacteria in its system than a domestic, so if you do get bit, a doctor visit for antibiotics is going to be needed. Its not likely that it has rabies, because for anything that small to get rabies, it has to survive an attack from a larger animal, which is unlikely that it would survive.
As cruel as it might seem, I'd release him into the wild, or call a wildlife rehab center in your area. I don't mean to sound cruel, as I have rats as pets, but it is one, wild mouse, and the chances of its survival in the wild when its healthy are slim to none... let alone injured.
Just be careful of parasites.