Thank you for being part of the Wefugees community and sorry for the wait!
I can see that you already asked a related question here: https://www.wefugees.de/802493/antrag-auf-elterngeld
Just to be sure: You mentioned that you and your partner are not married yet? Do you have a Duldung that is the temporary suspension of deportation/ "Aussetzung der Abschiebung"?
Regarding your first question - if you can apply for Elterngeld - I found a document by the Federal Ministry of Labour and Social Affairs that you can download as a PDF: https://www.deutscher-verein.de/en/uploads/dateien-stab-internationales/social-security-at-a-glance-2016.pdf
What is mentioned here on page 10 is that "Mothers and fathers can claim parental benefit if they
- Look after and raise their children from birth themselves
- Do not do more than 30 hours’ paid work a week
- Live with their children in one household
- Have a place of residence or are ordinarily resident in Germany"
"Other foreign nationals can claim parental benefit if their stay is likely to be long-term based on the type of residence permit they hold and whether they are allowed to work"
To make sure whether or not your are eligible for Elterngeld, it helps if you contact parental allowance offices or local services in your area. You can find a list on page 11 of who is responsible for administering parental benefit in each state.
As for your second question, what my research has shown is that buying the car is less of the problem, but registering it under your name, having a valid driver's license, and obtaining insurance is essential.
I hope the links and information here help you to clarify your situation, but please feel free to reach out to us again if you have any more questions!
All the best,