In EPESI we use Roundcube as email client, where you'll receive and send messages. Your account has to support IMAP protocol. Then each email that is important for your business can be archived, what means that it'll be copied to EPESI database (with attachments) and you'll be able to link this email to any record, where E-mail tab is visible. By default it will be in contact/company with corresponding email address. There is no auto-copy to EPESI, because of SPAM, etc.
And with premium support - you've got it right. If you've bought our time, we can do anything you wish (and we can).
Regards,
Adam