In case anyone had the same issue, here is how it was resolved: by installing the "November Update" for Windows 10 (aka version 1511). This was really driving us mad.. we`ve updated multiple machines from Windows 7 and 8 to 10 and then Outlook synced and received emails but could not send - all sent items would get stuck in Outbox (note: only occurred with EAS). What was even weirder, is the fact that for some machines this happened and for some it did not.
Turns out MS deferred the November Update (1511) for 30 days for newly upgraded machines, so machines that were upgraded earlier got the update and Outlook did not have this issue. The ones that were just upgraded, did not see the update and emails got stuck in Outlook`s Outbox.
So, this issue had nothing to do with SmarterMail, but with Windows 10. If anyone else is experiencing this, I hope this info will help.
@ Admins: how do I mark the thread as solved?