Outlook 2013 Not Responding and Offline Address Book Issues with Office 365

Question

Around 3 months ago I setup a small business with Office 365 for email. They have 10 PCs, all running newly installed Windows 7 Professional 64-bit, eight with Office 2013 and two with Office 2010. All are connected to a local domain (Server Essentials
2012) with a single internet connection through a new TP-LINK router. Each user logs into their own PC only.
The network runs perfectly, internet connection is good but there seems to be RANDOM, INTERMITTENT issues with email not being SENT from Outlook every so often. Email will sit in the outbox for up to an hour sometimes before, eventually, being sent. Incoming
email seems ok, although this is more difficult to tell. At others times, email will be sent instantly - the size of the message and any attachments makes no difference.
The issue has happened at some point on every one of the PCs but no particular PC/user seems worse than any other. When email is stuck on one user's PC it can still be sent ok by other users at the same time - so the internet connection is not down. Users can
can always login to the OWA service ok and iPhones etc. all sync ok even when email is stuck in the Outbox.
To debug the issue I have worked on one particular user's PC and carried out the following changes - 
Changes that made no difference
I have checked the router. A SPI firewall is available on the router but this is switched off.
I have changed the MTU on the router with no effect.
There are no unsual add-ins to Outlook
Outgoing email virus scanning is disabled
Outlook is NOT integrated with Lync
The OST file is around 1.7 GBytes is size.
I have recreated the Outlook profile twice but this made no difference
Switched OFF "Cached Exchange Mode" - this made things worse !
The Outlook address book is set to \Offline Global Address List
I have disabled with Windows firewall on the client PC - no change
I have adjusted the KeepAliveTime for packets in the registry of the PC to 20 minutes
Changes that reduced the frequency of "Not Responding"
I have changed the Outlook automatic Send/Receive time from 30 to 10 minutes - this seemed to help 
Possible Related Problem
I have attempted to download the offline address book manually - this hangs at "Copying offline address book template file" in the Send/Receive dialog and there a NO OAB files in the data file location on the PC. 
So the only measure that has so far "helped" has been the Outlook Send/Receive time change - now set to 10 minutes. Even so, Outlook will occasionally go "Not Responding" for 30 minutes or more (no windows active - only option
is to close and reopen Outlook).
There does seem to be a problem with the Offline Address book. I can download this ok for this user on my home PC (Outlook 2010) but this will not download manually on the work pc with Outlook 2013 (despite following all the debug steps above).
On work PC I can see literally hundreds of "Schannel" errors in the Windows System event logs - 
Error - Event 36888 Schannel
The following fatal error was generated :10
The internal error state is 10
These errors occur every 2-4 minutes continually all the time Outlook is running. When Outlook is closed the "Schannel" errors stop !
Can anyone suggest what else to try, please ?
I have already investigated this issue on the Office 365 Community Forum, and the Outlook forum and they suggested that it was a client (i.e Outlook) issue and that I should post here for further information.
Thanks for reading !

Answer 1

An update to this problem.
I brought one of the office PCs back to my home network and connected it up there. With no changes to the PC I was immediately able to download the Offline Address Book without errors.
This suggests to me that there must be some problem with the internet connection or router setup in the office environment.
The router is a TP-LINK W8970. The internet connection is PPPoA. I have seen some suggestion that an SPI firewall can cause issues. This router has an SPI firewall but this feature is disabled on the router.
While investigating this issue I have frequently been connected remotely to an office PC - sometimes for up to an hour or more. During these periods I have not experienced any connection dropouts so the actual internet connection seems to be ok (i.e. it is
not dropping out). 
However, something seems to be preventing the Outlook client from connecting properly with the Office 365 Exchange server in the office environment - or something is dropping the connection to Outlook. Does anyone know of any router settings that might affect
this ?
I could try changing the office router but I do not have a spare available. Can anyone suggest a reasonably low cost, currenbtly available router that they know definitely works in the Office 365 environment so that I could buy this for test purposes ?