@ The Letter
You can work without an NI number for at least a couple of months. It will mean that you are taxed at a basic rate of around 25% without any tax free allowances. Depending on your gross income it probably means you'll overpay tax for the first few months but as soon as your NI number is sorted you can write the tax office connected to your company a letter which will result in a nice big cheque coming your way. Obviously if you earn more then a certain amount you'll be underpaying and owe them something, cut off is around 35000 I believe.
I used the company head office address of my previous employer to set up a bank account etc, as long as its an address the bank is happy. They just have to tick boxes and don't really care as long as the boxes are ticked, well so it seems anyway.
@ Bleeds off:
No, as long as you have an EU passport you can live and work anywhere in Europe without a workpermit etc.