Firstly; you need to understand the difference between IMAP and POP3. POP3 is the default with BT email (ie from btconnect servers usually with @btconnect.com email adresses). IMAP is only available with the added Organise and Share service which costs £5 a month per email address service. You cannot use IMAP with BT unless you subscribe to their ridiculously overpriced Organise and Share service.
POP3 in its standard configuration sends received emails from the receiving servers (BT's in this case) to the addressee's PC or other device when they are read and removes them when it does so from the server. (That's why you don't see an email you've read on your phone or tablet on your PC when you get home).
POP3 can allow received emails to be seen on multiple devices (by ticking the Don't Delete from Server boxes in your email client settings) but you run the risk of running out of space on the BT servers. There is, however, no way you can see sent emails on multiple devices with POP3; they only ever exist on the device that sent them. There are other differences between IMAP and POP3 but these are the crucial ones in this case.
Secondly; BT do not use authentication on the outgoing (SMTP) part of their email service. You must therefore remove any entry in the username and password fields in the outgoing settings AND make sure the authenticate box is unticked AND stays unticked when you save the settings.
The simplest and cheapest solution - if you can bear to give up your @btconnect.com email addresses - is to move to Google mail, Hotmail, Yahoo or one of the other email service providers. These all offer IMAP as default, don't charge for their service and are much more suited, or in tune with, today's email environment where so many of us now have multiple email devices.
Good luck.
