Having weird symptoms with 5D MK II

Hello, I recently bought an extremely beat up 5D MK II for a price I just couldn't refuse. It was missing a battery door, rubber eyepiece, etc, but I had a feeling the camera still worked so I picked it up.

Fast forward a few days and I now have the camera. I slap in a battery and try to turn it on. Seemingly the camera doesn't do anything, but the top display and viewfinder lit up. At first I assumed the rear display was just toast, but it wasn't shooting. I hooked up the mini HDMI output to my TV, but still nothing. I could change iso, shutterspeed, etc... so I was confused at this point.

Then suddenly the camera just... worked? Rear display still wasn't outputting but I could change the settings on the tv, take pictures. Shutter seemed to work fine and sounded great. Then I turned the camera off to put in a card to test picture output and now it won't turn back on fully, just outputting in the viewfinder and top mounted display lol...

I'm no Canon user, I shoot Pentax. I'm really not familiar with the possible faults on these cameras. I was hoping you guys could help me if you've heard of similar symptoms. Clearly the camera isn't just fully dead, even if the rear display is toast I don't really need it to take photos with.

Thanks for your time.