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, 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.)
- If the plunge pool is in the sauna area, then you should wear the same attire you would in the sauna. (If you wear nothing in the sauna, then wear nothing in the plunge pool.)
- Any number of people can use the plunge pool at once, if they fit comfortably. Don't block the entrance/exit.
- You may stay in the plunge pool as long as you can comfortably do so, provided you're not blocking other people who want to use it.
Obviously this means that for a single-person plunge pool, you should go in, go under (if desired), and get right back out.
For a larger plunge pool, hang out and freeze if you like; just make sure you're not in the way of others who want to use it.
As far as the "tap that fills the pool", I'm not sure what this is. Around here, plunge pools are filled and filtered automatically.