my hamster got wet tail but now?

the vet said they normally don't live more than 24 hours and I said NO he will live and took him home and now it is day 6 and he's going strong.
I was wondering if it was safe to give him a tiny careful bath today to clean him up I've been terrified to touch him or disturb him all week in case it makes him worse.

1. Will his runny bum have stopped by now?
2. Is he contagious with E-Coli to humans?
3. Should I wash him gently and pop him in a smaller cage with paper so I can see if he's pooping again.
4. This must mean he's eating and drinking something to keep going, his cage has NO ladders or anything so it's ALL climbing and he has been moving around that cage climbing through his tubes, into houses, ground floor etc and I mean stretching and climbing it's an exercise cage since last Sunday. I've been putting food in but I don't think he's been eating loads of it but he must be drinking too.

He's been separated from the others since last week and put into our garage where the temperature is kept pretty good. He doesn't open his eyes that much but I think he's starting to feel better now. Will his diarrhoea have stopped by now? I've been a good nurse to him.