Nikon D200 battery door - is it broken?

The battery door is meant to come off so the MB-D200 battery pack can be inserted, but it sounds as if yours has broken. There should be equally-sized rods that form the pivoting hinge, on on top and one on the bottom. Yes, you can buy a replacement part, probably through a local dealer or online. I believe online they cost around $20-$25. To remove the door, you are supposed to open the door to a 60-degree angle to the camera body and gently pull the door off. To replace the door, again, hold the door at a 60-degree angle to the body and gently press it into place. Or you could pay a repair shop an arm and leg to do this for you ($50 or more + shipping if you use a mail order repair shop).

I remember the 'good old days' when film SLR's were made to take a beating. Sadlly, we're in the era of plastic and I always have to remember to treat every piece of equipment as if it were made of glass!

Hope this helps?