I'm not sure whether the rules differ in Hungary, but it's probably the same as in Germany and Switzerland:
If you're sweaty, you should always shower before entering the plunge pool. (You don't have to observe this if the "plunge pool" is a natural body of water such as a lake.)
Any number of people can use the plunge pool at once, if they fit comfortably....
Nothing, you can keep it as a memento.
Last year I travelled to Singapore with my friends, our return flights were different and I was the one who checked out last from the hotel. When the receptionist asked me about the other 2 cards, I told her that we lost them somewhere. She said no problem and wished me a happy journey.