Uncle had a skunk get into the drinking water well and decompose. Smell started pretty quick, no one died.
No one? Doesn't sound as if the SKUNK - North America's primary reservoir for the rabies virus - lived to brag about it to the grandkids?
As to mice, rats, and tree-rats:
Diseases directly transmitted by rodents | Rodents | CDC
There's a link in the sidebar as to cleaning-up.
Grandparents did not HAVE the gadzillion of packaged chemicals we have in "Big Box" for scores of years.
They did their work with "BFBI" methods. HOT water with strong brown soap made on the farm from saved-up wood fire ashes, vinegar made from our own cider, salt, lime & quicklime, now and then coal-oil, and store-bought Bromine, Iodine, Ammonia, Clorox, Potassium Permanganate, even Zinc, Copper, Sulfur, Mercury, Tin compounds, Nicotinic Acid, and Arsenate of Lead.
Kinda careless by modern standards. Local stores selling blasting powder and dynamite as well.
Used carefully "enough" that most lived into their 80's, 90's, now and then past 100.