Why do I keep getting alerts to update, but, when I check, I'm told I'm using the latest version of FF?

Your System Details list shows that you use a Firefox 31.0 release. That is not the current release. The current release is Firefox 34.0.5.

If there are problems with updating or with the permissions then best is to download the full version and trash the currently installed version to do a clean install of the new version.

Download a new copy of the Firefox application and save the disk image file to the desktop

• Firefox 34.0: https://www.mozilla.org/en-US/firefox/all/

• Trash the current Firefox application (open the Applications folder in the Finder and drag the Firefox application to the Trash) to do a clean (re)install
• Install the new version that you have downloaded
• https://support.mozilla.org/kb/Installing+Firefox+on+Mac

Your personal data is stored elsewhere in the Firefox profile folder, so you won't lose your bookmarks and other personal data when you uninstall and (re)install Firefox.

• http://kb.mozillazine.org/Profile_folder_-_Firefox

When I drag the old FF to the trash, a message scolds me that the files "cannot be deleted".

You must install the new Firefox. This will remove the old program for you.