-So far in 7423 - Been running for an hour or so, and I have not been prompted for my password at all.  Much better than 7419 which was every 5 minutes.  Not sure what changed, but this is good.  and my connection remains stable.

Nothing has changed in regards to how we authenticate with MAPI.  If for some reason there are network problems between the client and server you will get authentication prompts.  For instance, if the client cannot reach the server for a period of time during a MAPI request then the authentication window displays and the client will stop until the credentials are entered.

-What tells MAPI to sync and when?

From Outlook there's a manual sync where you can click the 'Update Folder' button under the 'Send / Receive' tab.  If there are changes on the server that the client does not have clicking this button should always pull the new changes.

From SmarterMail we send a notification to the client letting them know that a folder has some changes and that prompts the client to sync that folder.  This is the automated process.

-When I delete from MAPI in Outlook, it never seems to really delete the message.

This should never be the case for a normal mailbox.  This usually means that the client didn't like some information it was given and so the folder will not update (it will not sync new data from the server or push data to the server).  If a client gets into this state then it will affect the issue you brought up where your client is out of sync when compared to other clients. We have fixes in build 7423 that should have address some of these issues.  If you haven't done so already, delete your profile, delete your Outlook .ost file, remove MAPI from your connectivity tab in webmail, then resync your entire profile.

-And I have already alluded to this above, but is MAPI just slow?

I'll let others in the community answer this question as we've not seen slowness in our testing.  Just to provide some insight, the basic architecture is similar to that of EWS.  So in theory, it should be similar speeds from a web request/response time frame (not from a syncing time frame as both clients sync data differently).

Is there anywhere to see the MAPI sync progress either in Outlook or in Smartermail?

If Outlook is currently syncing a folder, then at the very bottom it will show a message like 'Updating Inbox'.  SmarterMail does not currently have a way to show this.   The other way to know if its syncing is to use Fiddler to sniff the web traffic but it's not easy to read the data.

My experience is currently

1st a Microsoft Sign in Form appears, which needs to be closed by the user

2nd Connection lost appears

3rd Click on the Need Password and continue

Is it likely we have something incorrect in our setup

If I delete an email or even a series of emails from a folder, webmail updates within ~10 seconds, my iPad and iPhone which use EAS update as well, within ~10 seconds. 

Much appreciated, as always

Per my previous response, I tested with an account connected to an actual Exchange server, OWA was updated within ~3 seconds and my iPad over EAS updated around the same time, ~3 seconds.

This is an internal Exchange server that doesnt really do anything, so comparing it's time to SmarterMail isnt fair considering our SM install is loaded with many many thousands of connections and users. Given that, I'd say the 8 - 10 seconds it took SM to update and send out the push alerts to my devices isnt too bad :)

I agree, today has been better.  Still quirky stuff going on.  But could be partially on my side.  Getting the message that Windows Defender is not configured properly, however we are using SCEP.  Have added the same rules.  But not getting proper results.