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.
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.