Mail does not work properly with iCloud on 5K Retina iMac 27" late 2014

I don't know how to further clarify my question - apologies. I am working on 10.11.3 (15D21) and this problem has happened over the last few updates of El Capitan and is getting worse.

When I open Mail and move emails from my Inbox to Trash, or Archive, or Junk or another dependent mailbox, the changes do not stick. So if I close Mail and open it again, the emails have not moved. Or if I look on iCloud, the emails I moved in Mail are still in the Inbox. If I send an email, the 'whoosh' sound will be heard, but the mail will not have been sent. If I'm lucky, a Draft might appear (and sometimes several drafts), but otherwise I get no notice except that the email is not in 'Sent' on iCloud. This has been going on for a few months and getting worse. It's like Mail is not communicating properly with iCloud mail servers at IMAP and SMTP level. On my other versions of Mail, I often have a little notification lower left corner saying 'downloading messages' which may show a few hundred if I haven't opened Mail for a while but it shows progress and completes. But in the bad version of Mail, it might say 'Downloading 11 messages) and any progress or complete takes forever or does not happen.

Now here are some qualifying factors which are important:

1. I have a MacBook, a separate installation of El Capitan running on an external drive on the Retina iMac, two iPads and an iPhone. These issues only happen on this one instance of my Retina iMac (so I guess it must be machine specific software). I have no problems when I use iCloud on a browser.

2. I have tried deleting all files starting with Envelope in ~/Library/Mail/ changed these files names, put them in Trash, emptied Trash then restarted Mail and my iMac. The entire mailboxes were rebuild but the problem quickly returned.

3. I have tried rebuilding each Mailbox using the Rebuild under Mailbox within Mail.

4. Automatically detect and maintain settings is always checked in the iCloud preferences.

5. This does not effect the second mail account I have in Mail (a separate IMAP account with a different mail server) although, because this is driving me nuts, the last time i rebuild the mailbox I did not add this account back.

6. I have followed the steps on online articles to totally rebuild Mail by removing all instances of ~Library/Mail and ~Library/Application Support/Mail and ~Library/Containers/com.apple.mail etc and ~Library/Preferences/anything with mail Plist etc then emptied the Trash then restarted my Mac then allowed the build to go on for a day or so but the problem recurs.

Any recommendations on how to completely reinstall Mail would be hugely appreciated because I am obviously not doing a complete job and there is some faulty connection being remembered or a corrupted index file somewhere.

With thanks, John