My friend told me I need to take my salary list to do with the register. I'm not sure if it is the "FORMS FROM YOUR EMPLOYER TO PROVE YOUR ELIGIBILITY"?
No, it's not the same thing. There are specific forms that your employer must give you (to prove ZUS payment) - proof of salary isn't enough, you need these specific forms. Or if you have the ZUS book, then a stamp on there is enough.
If I tell the doctor I don't feel well, will they have me checked right away, or I still need to make appointment and wait for a long time?
Depends very, very much on the individual practice. You're free to go to any doctor, though in my limited experience, you usually need to call or visit the practice first thing in the morning to get an appointment for the same day.