Welcome to the Wefugees community. I'm very sorry to hear about your situation, thank you for sharing it with us.
Let's have a look at your case and questions separately:
- You have applied for asylum in Italy and have been rejected
- You have a chronic illness but lack the necessary medical support
- You would like to move to Germany
I found a few sources that might be helpful in your case.
1. Regarding your first question, I found this website: http://w2eu.info/italy.en/articles/italy-asylum.en.html
Since you received a rejection, here's information on what you can do from the link above:
"5.10 What if you receive a denial (diniego)?
If your application for international protection is declined and you receive a denial by the territorial Commission, you can contact a lawyer to appeal to the Court. The court which hears your appeal is the one located in the capital of the District of the Court of appeal in which the territorial Commission is established or the SPAR if you are in such a centre. From the date of refusal by the Commission, you have 30 days to file an appeal, upon expiry of which you no longer have the right to stay in Italy legally.
If you are in a “CIE” or “CARA” you have 15 days to file an appeal.
Please refer to the operators of your reception centre or at the organisations close to where you are (see contacts) to find a lawyer to take your appeal. You can ask an Italian State aid called "legal aid" i.e.. the possibility to appeal against the refusal free of charge if you have an income less than approximately 11,000 euro."
2. Concerning your chronic illness, it is definitely important to take action to prevent any further harm to you. When I was researching health care in Italy for asylum seekers, I found this: http://www.asylumineurope.org/reports/country/italy/reception-conditions/health-care
As it states on the website, asylum seekers must sign up with the National Health Service, but you have the same right to health care as Italian citizens. Since the right to health is considered a fundamental human right in Italy, "[Asylum seekers] enjoy equal treatment and full equality of rights and obligations with Italian citizens regarding the mandatory contributory assistance provided by the National Health Service in Italy."
Here's also a guide to health care in Italy: http://www.salute.gov.it/imgs/C_17_opuscoliPoster_128_allegato.pdf
The only thing I'm unsure about is what that means for you as an asylum seeker who has been rejected. However, the PDF guide might help to get some answers. There will be a Local Health Authority (they call it ASL) in your area that you can contact as well.
3. Regarding moving to Germany: Since the Dublin III Regulations, you have to apply for asylum in the first EU-country that you arrive in. Meaning - if you do come to Germany, you might be send back to Italy, as Italy would be recognized as the first country (since you applied for asylum there). Italy would therefore be considered responsible for your asylum case. See this Wefugees thread for a similar question: https://www.wefugees.de/102218/can-rejected-asylum-seekers-in-france-apply-again
Maybe our asylum law expert @Nilab can add anything concerning your specific case.
For more information regarding your questions, it can be helpful to contact an NGO supporting asylum seekers in Italy, also with appeal cases (here's a list: http://www.refugeelegalaidinformation.org/italy-pro-bono-directory).
I hope that you will receive the medical care you need soon. Please let us know if you need any further assistance!
All the best,