if you lived in a second country after your home country, the BAMF will check, if it is possible for you to return to both of them. For example, if you had a residence permit in the second country which was not dangerous for you, they can check if it is still valid or can be prolonged or if this country can take you back. If both countries were dangerous, you must give proves for that. Sometimes BAMF just rejects people because they were safe in their second country and threatens to send them there without proper checking if it is possible. In this case you can have an argument for the appeal to the court. So you must be prepared to explain in your interview why you can' return to both countries.
You can propose to show physical proves if is ok for you. It is anyway advisable to have an attest from the doctor about the seriousness and the character of the problems which can strengthen your story.
It is not so easy to move to another EU-country after gettig asylum because they are not obliged to give you social help. If you find a job and can earn your living, it is often possible, but you must check the conditions for getting a working permission and a long-time residence permit in each special country. Normally you must apply for that in the embassy of your aim country. Only after 5 years with a residence permit and after fulfilling some condition (for example earning money for some time) you can apply for a special EU-wide longtime residence permit with which you can live and work in each EU country (but social help is also difficult to get here).
After getting some form of protection and a corresponding residence permit you can travel outside EU, but you will need a visum if the aim country is not visum-free for you as a citizen of your country of origin. And you naturally need a passport. If you get a refugee status, you will have a refugee passport, with other forms of protection you normally need your national passport.
You have many questions and the answers have too many aspects. Therefore it is important that you seek for a good councelling center for migrants in your area!!! I only gave you general information.