Advice on living in Czech and working in Germany?

I currently have a full time professional job in Germany but I plan to move in with my Girlfriend in the Czech republic. I hope that someone is able to help with the list of questions I have below: 1. Do I need to register with the police in Czech to state I am living here? 2. Will I still continue to pay my income tax and insurances to the German Government? 3. Will I be allowed to keep my German bank account etc? Is there anything else I should consider? Many thanks Chris