Welcome to the Wefugees Community and thank you for reaching out to us.
As you are an EU-citizen, once you and your fiancé got married, EU/freedom of movement law will apply. If you as EU-citizen fulfill the requirements for the freedom of movement, your then husband will also be treated in the same way and has the legal right to stay. Please follow the link for more details on this. Unfortunately, I am not an expert on EU law, but I think if your fiancé will base his residence on the fact of being a father of an EU-citizen, the situation is a bit more complicated. I will link two of our experts here. Dear @mbeon-fardeen or @mbeon-Ruth, can you give us maybe some more information on this?
In any case, officially declaring the fatherhood will be the essential first step. Applying for a residence permit according to Section 25 (5) of the German Residence Law might be an option for the time being. However, I doubt that the immigration office is obliged to issue it. Maybe it can be helpful to get support by a lawyer in this matter.
Regarding the rental contract: If you are talking about renting a flat, this is also possible with holding a Duldung. At the end, it depends on the landlord's/ the company's preferences and choice. In some cases it might be easier for you as EU-citizen to sign an contract.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,