Welcome to the Wefugees community. I'm very sorry to hear about your situation.
To answer your first question: You have the right to apply for asylum in Germany.
If you haven't applied for asylum in any other EU-country, and if no other EU-country issued you a visa/ your fingerprints were taken, then the Dublin-III Regulations wouldn't apply to your case.
Unfortunately, it is impossible for me to say if you will receive a positive decision on your case. However, once you applied, you can seek legal help from refugee support centers or a lawyer to understand what options you have. They are able to have a much closer **** at your case than we are here. Here is a list: https://***.proasyl.de/en/our-network/
If you are wondering how the asylum procedure works in Germany, have a **** at this BAMF website (Federal Office for Migration and Refugees): ****://***.bamf.de/EN/Fluechtlingsschutz/AblaufAsylv/ablauf-des-asylverfahrens-node.html
Please have a **** at some threads from our platform where people have answered similar questions that can be helpful:
What are the steps to take in order to apply for asylum in Germany?
My Schengen visa expired - Can I claim asylum in Germany?
How to apply for aslyum?
Regarding your second question: Other ways to stay in Germany legally are, for example, through taking part in a vocational training/ studying at a university/ taking up employment and applying for the fitting residence permit. Other than that, if you are married to a EU-citizen (German), you may live with your spouse.
I hope this information helps. To find out more about residence permits or work visa, simply use the search bar on the right! There is a lot of helpful information on here :-)
Feel free to reach out to us again if you need any further assistance.
All the best,