Rabies can be transmitted via organ and tissue transplant, so I don't think that's correct, but I'm no guru.
I don't believe that the rabies virus can survive after being cooked at a high enough temperature for a decent length of time.
But then again, maybe I'm wrong about that.
I'm not sure if there's a "field expedient" method for determining if the carcass was infected by a disease such as rabies or tuleremia, other than by simply taking note of the warning signs that Taurus brought up (foam at the mouth, bloating, horrible smell, etc.). I suppose that all you can do is to cook the meat thoroughly and then hope that the heat is going to kill whatever microscopic organisms may be lingering in the critter's carcass post mortem.