Hi,
I am still not sure from the 10.33 post (the local time I see) whether you have just the VPN or whether there is still another Internet connection going on ?
The Issue in Leopard was if a Leopard computer had Internet Sharing turned ON then the computer that was doing the Sharing could see both the Internet connection it was given but Failed chats in iChat also reported the Originating Mac's IP address.
Moving to your Last Post there is Location checks.
iMessages Account is set to your Location.
in Mavericks and Yosemite that is set in the Details button in Messages > Preferences > Accounts > iMessages account and the Details Button.
You have to Sign In here to view the details.
AS a security check this detail is checked against the IP that your location appears to be at.
AS the Location and IP's location may not match then the iMessages server will not let you login. (you look like someone pretending to be you).
It also messes with Time Stamps (and the Login is Time Stamped) which means even if you do manage to Login you may miss some iMessages because of time differences between the VPN's Location and the Mac's.
If the iMessages servers see this sort of behaviour then it will block the Apple ID for that Mac.
That is to say the iPhone should still see sync copies of iMessages sent to the Apple ID but the Mac will not.
Nor can the Mac send as it cannot login.
The only way to resolve that is to Contact Apple Support and move up to a Level 2 person and ask them to reset it.
However it is not that simple as the Upgrade from Yosemite to El Capitan is also triggering this from some people and the Level 2 people involve the Engineers who are looking for the Trigger.
You have an Apple ID issue which are free to fix. (Level 1 people are Script Led looking to solve Apple Care Hardware and Software issues).

8:17 pm Tuesday; November 17, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad