I'm a Peace Corps volunteer and wanted to download my library to an external drive so I could install the games I wanted to play during my service so I don't clutter my laptop with games. When I went to install some games I only managed to install three (Duke Nukem 3D, System Shock 2, Fallout) before all the other installers stopped working and the icons changed to the default .exe icon in Windows 10. Now I just get an error that says this program can't run and no compatibility settings or changes seem to get them to work.

Can you run other .exes?
This definitely looks like a Windows problem, not a GOG problem. And oddly enough, just checked AskWoody and it's the title issue in the posted newsletter now, see https://www.askwoody.com/askwoody-newsletter-alerts/what-to-do-when-windows-cant-run-exes/
Can also be some security issue, from the installed security software or some other implemented security feature, especially considering your situation, if some sys admin tinkered with that laptop. In such a case, may be that the GOG signature was flagged as suspicious for some reason, or even that games were flagged as undesirable. May also apply to that external drive you use.

Maybe your external drive is not connected properly? Usb cable wonky? If the icons revert to Windows icons it means that they cannot be read from the drive.

Can you run the installer files after copying them from the external to internal drive? Access anything else on it, eg, photo's, music, video? I'd definitely check the drive first.