If you move from Germany to another EU-country you have the right to receive medical care in the new country.
It is unclear from your post whether you intend to move around for short or long periods of time, and so I cannot give you very precise information.
But most EU-countries require that you register as a resident after 1 year. As a resident you then get access to health care in the same way as everyone else.
If you only plan to stay for less than a year, then different conditions apply. The medical care you are entitled to depends on what kind of medical insurance you have in your country of origin, i.e. Germany.
As far as disability is concerned you would potentially receive disability from Germany, not the new EU-country if you are not a resident in the new country. I say "potentially" because you would have to ask your health insurance provider if they allow you to cash in disability from Germany while living in another country.
If you become a resident in the new EU-country you can apply for disability there.
As far as moving around is concerned, you are free to move to whichever country you want. The duration of your stay in one country determines whether you must notify the authorities in your country of origin and apply for residenceship.
