Yes and possibly.
Anything that crawls, walks, runs, swims, or flies is edible. I have been is some pretty remote parts of the world, and you would be amazed at what people will eat, especially when they are hungry. That is why I call the animal rights movement a rich man's sport.
The vast majority of diseases can be cooked out of meat. The only ones I can think of that cannot are the transmissible spongiform encephalopathies, the most common of which results in Creutzfeldt-Jakob disease. Fortunately, there are no known cases of rodents transmitting this disease via ingestion.