Can't see a reason why you can't do that. Disconnect old battery put the new one in, be careful though as you won't be able to put back cover on. You don't want battery flying around and taking the logic board chunk with it. Just out of interest, did you check whether your model was in a battery recall programme? If it was you should have had it replaced by Apple for free 15-inch MacBook Pro Battery Recall Program
Yes, it sounds very much like a battery issue. I am guessing it is not a Genuine Apple battery. Try to drain it completely before charging it up again to full, but I suspect only an original battery will take that error away in diagnostics.
If you can see the image when you shine the light on the logo at the back then it probably the backlight on the screen. You do need a new screen assembly and it's quite easy to fit. To output to external monitor as your primary monitor start your Mac and when it boots up then close the lid. The following guide should help: http://osxdaily.com/2010/12/21/broken-macbook-pro-screen-turn-it-into-a-desktop-mac/
I had the same issue with a 2015 Imac and I tried fixing by creating a boot drive but whatever I did, mac wouldn’t see it as a bootable drive. I resolved it by taking the hard drive out of the none working mac, connecting it as an external drive to a working mac and cloning the drive using Carbon Copy Cloner software, that fixed my problem. You should be able to achieve the same by doing Target Disk Mode, but I did not have a Thunderbolt cable at the time.