Hi,
iCloud is nothing to do with iMessages.
In fact in System Preferences > iCloud there is nothing for the iMessages account, the Messages app, FaceTime, iTunes, App Store, iBooks, Game Center.
Realistically there is no difference is using a Google mail ID (or mail ID of your choice) as an Apple ID compared to an iCloud ID used to login to various Apple servers.
Reset Home Folder permissions and ACLs Error
There are two important posts in this thread.
Do the Orange/Helpful one first. It tests if you Own and have access to all the files in your Home Folder including the Library and Keychain items.
It also checks you have the correct Read and Write permissions.
When you have run the test you should get a number (Linc suggests 41 as an example).
Whatever the number (you will always get some) run the Terminal Instructions in the Solved/Green Post.
The reason I am suggesting this is that the Preferences files (.plists) are held, in the main in ~/Library/Preferences
However to Write to some of them you have to go ~/Library/Containers/com.apple.iChat/Data/Library/Preferences that then have Aliases pointing to the file sin the other folder.
This dramatically increases the number of Folders you have to have Read and Write permissions for.
Manually checking and resetting each one (potentially) is laborious.
The above Link will get us past that stage without missing anything.
Once you have done this then restart Messages and try the iMessages account again.
Report back at this stage.
I am likely to go off line in about 1/2 hour and it will be about a 20 hour wait before I am back (I will be back) (Sleep, Work, Eat -> Online)

9:29 pm Wednesday; August 24, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone and an iPad (2)