We do use WPEngine, so I was able to contact them to have the cache exclusions added as described in your support article, which seems to have resolved issue 1.
I had to wait until this point to check the JS Console for issue 2 again, as we had rolled back the site in the meantime. Now that issue 1 is resolved, I’m unable to reproduce the second issue with the Save Changes button. Unless it could have been related to caching as well, I have no clue what happened there.
At any rate, all our problems have gone away. Thanks for your help with this!
