I installed the backup locally and after jumping through a few hoops to be able to access the admin pages I tried to see the problem but could not, I had no problems switching back and forth between the visual and text tabs on the post edit screen or editing a date custom field, and saw no errors in the console.
Except!
I installed it on a current WordPress installation, i.e. WP 5.0.1.
There was a database upgrade notice, so I suspect you were still on 4.9.9, is that right?
I installed the Classic Editor plugin so that the edit screens are the same as with 4.9.x.
Given that I cannot reproduce the error in that case, is it an option for you to upgrade?
I did the upgrade to WordPress 5.0.2, but that did not change anything regarding the problems.
I will do some more updates of plugins and will get back
After updating Toolset and All-in-One WP Migration things changed.
I can now use the Visual/Text-button, and also custom date and image fields.
That was a little unexpected but it looks like that solved the problems!
But perhaps the problems will return... but no need to worry now?
Well, we haven't been able to establish what the actual cause was, so I can't be sure it won't reappear, but if everything is updated to the current versions and you are not seeing the problem then I'm hopeful that it has resolved itself.
If I leave this thread open awaiting your feedback then you can let me know if you see further problems over the next few days. (No need to reply now.)