Yes. The hamster's should be separated, the cage should be cleaned completely, and the hamster should receive immediate medical attention or else it may die.
Wet tail in a hamster is not transferable to a person. When a hamster has wet tail, the main symptom is diarrhea.
Well first of all there should not be mold in a hamsters cage. Try taking your hamster out of the cage and keep her/him in something else for a while. Scrub his/her cage really well, get hamster "cage clean" to get the stink out. Your hamster should be fine.
You need to clean out the cage once a week. You should not need to bathe your hamster (although I do once every few months). NEVER put anything in your hamster's anus - they poop it all out. If, however, your hamster's nether regions are wet, it may have wet tail (diarrhea) and you should rush to a pet store for a product called DryTail that works well or bring the hamster to a vet immediately.
It probably has a disease so you should take it to the vet and clean their cage
Try not to disturb your hamster and just dispose of soiled bedding but do not clean the cage until it's wet tail is cured. You have to take it to a vet or else it will die
it will get a really bad cold and be sick
Hamster cages should be cleaned by removing all bedding, toys, etc. Rinse out the water bottle and let it dry out. Once everything is out of the cage, take a damp cloth and wash the floor and sides of the cage. While you are cleaning the cage, you can put your hamster in the sink or another place that they cant get out of with your back turned.
Every 6 days, but its a good idea to remove wet woodshavings from wherever in the cage the hamster urinates every day. or once a week
Once the babies have become independent and are not feeding off the mother. If you handle the babies or put your scent in the cage before this time, the mother will eat the babies.
the pungent smell might be a condition called wet tail if so you should check the bottom of your hamster and if the back area is wet and covered with feces you should take your hamster immediately to the veterinarian.
DO NOT PUT A HAMSTER IN WATER! It could get sick and possibly die. If you want to bathe a hamster, go to the pet store and get chinchilla sand, not dust, and put a little pit in a bowl and place it in you hamsters cage then it will clean itself.