Battery and LCD panel are failing, can I operate it "headless" safely?

It should work fine, I attempted somthing similar with an iMac G4 17", and it worked out pretty neat. The computer operated normally, and still thought it was an iMac though. Note: the graphics will not be as powerful, as if they were through the computer's screen, since it is Mirroring video out of the DVI Port(Not a major decrease, but somthing to keep in mind, DVD viewing should be fine) , since it will still think it has the screen installed, and it may or may not show up as a white screen, instead of the Apple with the spinning wheel upon boot, but it will show up after Mac OS X Loads. This only happened the first time i booted the iMac though, after it realized there was no display connected, and changed the preferences. Other than that, it worked fine, and seemed safe. Laptops like this will require a 65W Power Supply to run solely on AC Power.

Absolutely! I had a similar machine, one with a failed display. I chose to remove the display + lid completely - hinges, cables, and all. It worked just fine: If you choose JUST to remove the LCD panel, and leave the frame and cables in place, for better cooling I'd leave off the "lid" and remove the magnet that's associated with the "lid closed" switch. Hope this helps!