Motherboard repair and OS
Would overheating when powered on, just sitting idle, be remedied by replacing motherboard? If so, do they come with OS pre-installed or would I need to flash OS?
How hot was it getting? Was it warm or too-hot-to-touch? Do you find that it charges properly? Does the battery last for a reasonable amount of time or does it die within an hour?
Android is installed on the internal NAND or ROM of the phone motherboard. If you buy just a motherboard replacement off ebay you may get one with android installed or you may get one that is blank and just has a bootloader. It depends on what the seller did to phone before he/she removed the motherboard.
If you are swapping a motherboard from a working phone and that working phone has android installed, than the resulting phone will have android installed since android was installed on the NAND of that motherboard.