I am swiss but I will attempt to answer this as good as possible:
#1 Storage You get a bank account. Done. Bank account don't get you a lot of intrest but it is a safe way of storing the money and you'll have access to it at any time.
#2 Bureaucracy Germany is a little more bureaucratic than switzerland but a simple bank transaction should be enough.
#3 Taxes The free sum "Freibetrag" which you can gift to your children is 20k EUR in germany so you are got to go. This means no taxes for you or your dad.
#4 Liability for parents in debt If your girlfriend doesn't accept the heritage she's fine. You are not liable for the debt of your parents. If she would accept the heritage she would also inherit the debt. So don't accept it and everything is fine.
#5 Getting a flat with 10K EUR Not gonna happen, at least not in germany. You will need to save more and get a mortage but 10k is a nice start, you may want to start investing your saved money because you have the 10k security cussion on the bank account.
#6 Opinion: Girlfriend who hates her mother This is an opinion. But a girl who hates her mother is a major red flag. She's has 50% of her mothers gens so even if she is right about hating her there is still the issue that she can at some point become her. So my advice: run.