The dark system theme I have set seems to make some applications look funny, with dark boxes where they should be light. Is there any way I can manually set a theme on a per-application basis?
Yes, actually, one of the preinstalled themes is activated. Normally you have to pick the option that says something along the lines of "don't use themes" inside Tools/Preferences/LibreOffice/Application_Colors or something (depends on the language). That would make the desktop theme apply properly over LibreOffice.
You might need to reboot LibreOffice to see the changes take effect.