Yes you need the proper version of macOS to run this system if you want to run Mac Apps. The highest version I would install to start with would be Sierra (10.12.x) as you will need to fully wipe the drive to rebuild the GUID/Journaled file system (HFS+) in the process the recovery partition will be created. Then later you can upgrade to High Sierra (10.13.x)which will convert the systems firmware to support the newer file system APFS. then using a shim service like [guide|168463] you can run still newer macOS's!
So using the working Mac create a bootable OS installer with Sierra onto a USB thumb drive so you have a working macOS to even try booting the system.
One of the risks in this series is trying to find replacement MXM GPU boards as they are getting quite hard to find that are still working as both AMD & Apple tried to hide a chip failure then to compound things people out stretched what this GPU could support for Apps without melting down! So you have less than a 50/50 chance getting a working board. The next hurdle is finding an alternative board that Apple supports with there OS drivers that limits you or you get a better board that only runs in compatibility mode not gaining the full ability of the better board.
And yes, you may need to fix the boards firmware so the system accepts it.
So with the proper RAM and a working drive (either a HDD or SSD in the primary drive bay) as well as the needed and working PRAM battery and the display fully connected and the created USB thumb drive connected. You should be able to boot the system getting into the onboard diagnostics using the startup ***D*** key. Given the fact the GPU is Blown you will need to run the system in Safe Mode to bypass the loading of the GPU driver.
[https://support.apple.com/en-us/102603|Mac startup key combinations]
The system board also offers a set of diagnostic LEDs. But you need to have the display connected via a set of extension cables or carefully holding the display away so you can spy them. Be very careful! As the cables are not vers long and when display drops the force tends to damage either the logic board connector's or the cables them selves.