Is your router the Netgear DNG1000 ? (It will be in small print on the underside of the box).
I had one and it had all the symptoms you describe. It was junk. I wasted hours trying to get it to work, but the model is faulty. On Amazon, most of the reviews are bad, and complain of similar problems that you are suffering.
Amazon.co.uk User Recommendation
Mine came from my ISP, Virgin. I emailed asking for a new router. They phoned me back and said they could only send out the same netgear model, which I definitely didn't want. A small threat that I would have to go to a new ISP and they offered to give me a £30 credit which I used buy a decent new AC model.