Thanks for the investigation. It sheds the light on the situation.
SugarOutfitters is the store for SugarCRM. Apparently SugarCRM and SuiteCRM moved away from each other significantly. This is why if you are using SuiteCRM it is recommended to use the plugin from SuiteCRM Store.
Such thing happens if you have aliases for the mailbox. Do you have any other email addresses associated with this mailbox? Better send reply to support@grinmark.com because this case is public.
Could you send (to support@grinmark.com) the Exchange email address you used to configure the plugin? We'll use it to find relevant logs and check them.
The plugin requires CRM server to be accessible from external network. Whitelist our server cloud,grinmark.com (144.76.117.213) if you have a firewall.
It saves, sometimes Sugar does not display and requires fixes. I recommend check the history pane first. If no emails there than it is a different problem.
Yes, you can move away from an email while it is synchronized. Speed mostly depends on Sugar server. Also if you do not want to automatically link an email to contacts, leads or accounts you can uncheck corresponding options in Settings. Unchecking speeds up archiving as well.
In the logs I see that email were archived. Why you think they are not in Sugar? Notice that they should not appear in Email module, they should be displayed in History panels of related Leads, Contacts, etc.
Please follow this article to find out the version
https://support.office.com/en-us/article/what-version-of-outlook-do-i-have-b3a9568c-edb5-42b9-9825-d48d82b2257c#ID0EAABAAA=macOS