There is a chance that the equation dialog box appear outside your screen space. To test this assumption, please reset your SolidWorks registry keys.
I believe you are on the right track. I reset the registry keys, and the dialog box does pop up.
Based on this, I know what is causing the problem - the dialog box issue occurrs after copying over settings from the settings manager, even if that backup is from a stock "reset" file that worked previously.
So it would seem the actual act of copying over settings is causing this issue.
Is there anyway to be able to copy settings and still use the Equations editor?
Copy settings copies registry keys. So the answer is yes, if you spend the time to read those keys and edit or delete the ones that are causing the problem.
I tend to stay away from using the Copy Settings Wizard between different versions of SolidWorks. It takes less to re-configure my toolbars than it takes to troubleshoot these kind of problems.
You can experiment with limiting Copy Settings to keyboard shortcuts only. Or just eliminate the toolbar option first and see if it works...