Font drop down stopped working

You should always check the application requirements, prior to upgrading your OS.

Older versions of Adobe Illustrator system requirements

You need to upgrade your Illustrator or downgrade your OS. You will likely uncover more problems with such a large gap between release date of your app & OS.

CS6 is not supported for Windows 10. With each automatic Windows 10 update that has come out CS6 has gone beyond being unsupported to being on the verge of extinction. For many this has meant that it has started working erratically and for others it has stopped working at all. You can try trashing preferences (instructions below) but since as you say it is happening on multiple machines that probably won't be of much use. The best suggestions if you want to continue with CS6 are to try running your PCs in Windows 7 compatibility mode or roll back the OS to pre Windows 10 or get used older PCs from eBay. Otherwise it is time to get a subscription to Creative Cloud and update your software. Remember, as new operating systems come out they do not make allowances for older programs.

To trash preferences on a PC:

You can try the quick way of resetting on a PC which is to hold down Ctrl + Alt + Shift when launching Illustrator and respond affirmatively when asked if you want to reset. There have been some recent reports that the window asking if you want to reset is not popping up but that the prefs are being reset anyway. If this works great but if it doesn’t you may have to manually delete them.

To do so:

On Windows 7 and above the preference files are hidden. To find them go to the Control Panel and open Folder Options and then click the View tab. Then select “Show hidden files and folders” or “Show hidden files, folders or drive options” in Advanced Settings. Then delete (or rename) the folder at the end of this path: C:\Users\[User Name]\AppData\Roaming\Adobe\Adobe Illustrator [version number]\ Settings\<Language>. Make sure that Illustrator is closed when you do this. When you relaunch the program it will create  new preference files and the program will be at its default settings.

The advantage of manually deleting preference files is that after you’ve reset up the program (make sure that no document window is open) to your liking, you can create copies of your personalized “mint” preference files (make sure that you quit the program before copying them—that finalizes your customization) and use them in the future to replace any corrupt versions you may need to delete.