I'm not sure as I just installed the copy of the site you shared on my local server and I can not see any issues.
I need a way to see the issue to know whats going on.
- Did you try to disable the cache?
- On your live site - Do you have intermediary post type?
- What if you try to disable all other plugins and switch to default theme?
- Do you see anything with server's error log?
1. I've got ez-cache there, but it's only for front-end. It doesn't cache admin
2. Yes, there is intermediary post type.
3. I can't do it on live site. This is why you took a backup, isn't it?
4. Don't have access to server logs.
I suggest you should create a staging site on the same server using the copy of the live site and share the access details as we can not work/debug on live site.
I have set the next reply to private which means only you and I have access to it.
It looks like a hosting issue though. I've copied the site to another server and it worked fine. It's not the only issue with this hosting. I've advised the client to move the site to another hosting...