For anyone else who might have this same snafu, this was what happened:
Asset caching had been enabled in Debugging and Error Handling I usually set this as disabled, but we have been working on website speed for a while, so I enabled it. For two weeks, I had no problem with it.
BUT the uploaded image files resource url which is usually [civicrm.files]/persist/contribute/dyn is set to go somewhere outside of civi.
With the asset caching set, it was looking for js and css files in [civicrm.files]/persist/contribute/dyn, which is where it would store those for caching. This meant NO menu as the crm-menu related things would be stored there. I have no idea why I didn't have this problem sooner.
However, there was also a little piece of javascript in a formatting field in the profile I was using for this event, which caused the screens and errors above.
So, two unrelated errors, I think. An extra two hours troubleshooting. And, as always with me, it was a PICNIC error (Problem in chair, not in computer). Hopefully, my late night troubleshooting will help someone else.