it took longer to answer this time because I asked our councelling lawyer. She confirmed my opinion.
Having a residence permit of some kind (Aufenthaltserlaubnis) ist a necessary condition, if you make an application for an unlimited permit (Niederlassungserlaubnis). That means, the Ausländerbehörde must issue the first, limited, residence permit (in your case, if you got a refugee status, it will be a residence permit after §25.2 Alien Law (Aufenthaltsgesetz) for 3 years. As soon as you get it, you can apply for Niederlassungserlaubnis, provided you fulfil all the other conditions (which seem to be fulfilled in your case).
One of the conditions of getting a Niederlassungserlaubnis is having a passport. And as you have no possibility to use your national passport, als a refugee from your country of origin, you first need a refugee passport, which you normally get together with the residence permit or in near time.
I would take an appointment with the Ausländerbehörde, give them the application form for the refugee residence permit and at the same time discuss with them the possibility of issuing the Niederlassungserlaubnis just after the "normal" residence permit. Perhaps they can make a note in your file and check the conditions for both together, then it can be quicker.
About the German citizenship: our lawyer said that for this application you must really have a residence permit for 8 years, the time in the asylum procedure won't be included in this counting. If you provide a certificate of the integration course, then it is 7 years, and if your German level is better that B1, it can be 6 years (§10 Citizenship Law - Staatsangehörigkeitsgesetz).
I would recommend you to discuss this question later with the Einbürgerungsbehörde (the authority which makes the citizenship procedure) in your place. You can also consult a lawyer who is specialised in this area - but this probably will cost you money.