Whenever I connect to certain SS13 servers, my "Use graphics hardware" setting soon resets to being on, which seems to prevent me from seeing the contents of popup windows.

Has anyone else had an issue like this before? I'm on the latest version of BYOND.
Some servers force it on because it's needed for their visual effects. Eventually I plan to phase out the software renderer altogether.

The real problem is whatever's borked on your system that's preventing you from seeing the browser window contents.