Hi,
If you have done all of this and had no success it is likely that your Apple ID for the Mac use only has been blocked at the iMessages Servers.
Each device using iMessages has a unique login to the servers which involves the Serial Number of the device.
On the Mac it creates the IDS: Apple ID-Auth Token seen in the Keychain (The Italic bit is the email ID you use).
It seems that on Macs only that a security feature that uses this check is triggered and the Mac is then blocked.
The only solution is to phone Apple Support and have it reset.
Currently this is where the issue is.
When the Serial Number was first used a Pop up was seen if it was missing (OS X 10.8.2 and Messages 7.0.1)
There have been some related issue that have been solved the same way but it also seems people upgrading to El Capitan are also seeing this.
Level One people at Apple Support are Script led looking to solve Apple Care Hardware and Software issues.
Apple ID issues (which this is) are free to fix. They may try to charge you for the phone call.
Ask to speak to a Level 2.
Even then they are likely to want you to test in various different ways.
Some people reporting back have been told to Reinstall (not needed).
Insist that they reset the Apple ID on the iMessages servers for the Mac's use.
They may involve the Engineers who still seem to be looking for the trigger.
They may tell you that it will take 24 hours to reset (normally an overnight wait).
We haven't found a set of words that make this go any faster.
9:16 pm Thursday; December 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