I added this to my wp-config.php file, as described above:
define(‘SCRIPT_DEBUG’, true);
and that solved the problem of no editing toolbars and not being able to edit widgets and so on.
I had already tried disabling all plugins, switching to the default theme, and all the usual stuff, and none of that worked. But after adding the line above, I restored all plugins and my custom theme and all worked again.
When I checked the Error Console in Firefox, I saw that most of the errors were coming from bad code in the Gravatar CSS files as well as other files I have no access to.