Unless the law's changed since I was last in Germany - which was several years ago, admittedly - everybody living in Germany is expected to register their address with the police, whether they're German or foreign; and I think the time limit is much shorter than three months, more like a week.
If you're a British citizen then you're entitled to work in Germany on the same basis as German citizens.
Your entitlement to health care is a bit more complex. You won't be eligible for the European Health Insurance Card (EHIC - replacement for the old Form E111) if you're moving to live in Germany permanently. I'm not sure exactly how you'd go about arranging health cover; the Allgemeine Ortskrankenkasse (AOK) would probably be the best place to start, but your girlfriend will be better placed to advise on that than I am!
Viel Glück!