Can I apply for a German and a British passport or not?
Yes you can. You have both German and British citizenship via Jus Sanguinis (right of blood). The UK allows dual citizenship so that isn’t a problem. Germany restricts dual citizenship but one of the cases where it is allowed is where the child gets 2 or more citizenship at birth which is what happened in your case. So go apply! Good Luck!
Yes you can apply for both passports and then hold both of them at the same time. My family is British but I was born in the United States. Since I was born in the states that makes me an American citizen even though I don’t have any relatives from there. I carry both an American and a British passport.