3. if something is a norm - like a gym pass, then most likely you will get it without asking.
2. as Sunny Girl wrote that a citizen (and probably person with residence permit) can register to job office and get free health insurance. But then you have to visit Job office regularly (1st after a month, later after 3 months) and be ready to accept the job which they find for you (few times you reject an offer and they'll remove you from their list and you'll loose free insurance). Other option for unemployed is to pay for public health insurance. It's 337zł per month.
But then I read that there is single surcharge for people who were not insured previously:
w okresie od 3 miesięcy do roku od wniesienia opłaty dodatkowej w wysokości 748,16 zł
w okresie powyżej 1 roku do 2 lat w wysokości 1 870,40 zł
w okresie powyżej 2 lat do 5 lat w wysokości 3 740,79 zł
w okresie powyżej 5 lat do 10 lat w wysokości 5 611,19 zł
w okresie powyżej 10 lat w wysokości 7 481,58 zł
So it seams cheaper to get insurance from Job office, at least for 1 month and then continue paying this 337zł without need to pay 7 481,58 zł surcharge for lack of NFZ insurance history.
1. Public is only one - NFZ, there are also private insurers, but there is obligation for employed people to pay some percentage of their salary for public one. So private health insurance can be only addition to public (in order not to wait in queues for simple treatments)