OK, it sounds like the system has a graphics card, and is not too old. I think the next step might be to try a live Linux disc or USB drive to determine if you have similar issues with that. The computer is not staying up for 24 hours at present, using the current Windows OS.
Running the computer from a live Linux disc or USB drive, you could still access the internet and check webmail. If you use other application software under Windows, that would not run, but a 24-36 hour test should not be too much of an inconvenience. I would leave it it running 24/7 for the test period. If it crashes or reboots, that would tend to indicate a hardware problem and it might be new computer time. If it runs fine, it might be time to try a reinstall of Windows 7 or an "upgrade" to Windows 10 Pro to see if that eliminates the BSOD problems. you would want to make an image backup of the C: drive before trying that.
