I have been using this version of PS for many years, and now when I open any image and simply click inside the image (with any tool) PS quits. Has anyone else had this issue? I'm worried it's a non-compatibility issue with this OS version.
Try this, open System Preferences> Security & Privacy>Privacy, click on the Padlock, enter your Password
now select Accessibility in the left hand column. In the right hand box click on the + button, now navigate to your Applications folder, click to highlight Adobe Photoshop CS5 and from the column that then opens select Adobe Photoshop CS5.app and click the Open button to add it to the list. Make sure the box is checked.
Now restart your Mac, open CS5 and see if this has resolved the problem.
The other thing you need to have installed is the legacy java 6 runtime for macOS, you can get it here,
https://support.apple.com/kb/DL1572?locale=en_US
Some people report they can get Photoshop CS5.1 to run and some don't, you have to remember CS5.1 is now very old you shouldn't be surprised if it doesn't work.
Is this installation of Mojave a new clean installation or did you upgrade over an existing copy of macOS.
You might be better served by posting a question on the macOS Mojave community on the Apple Support pages,
https://discussions.apple.com/community/mac_os/mojave