Hi,
Forget about the system Preferences > iCloud bit as it has nothing to do with Messages.
There are several servers that the iCloud pane cannot access: iTunes, App Store, FaceTime, Messages, Game Center, iBooks to name the most common.
In Messages Menu > Status the Offline will only refer to Buddy List accounts like Bonjour, Jabber, AIM or Yahoo.
It does not apply to the iMessages account.
Therefore disabling Bonjour and your Jabber account will still show you as Off Line whether the iMessages account is logged in or not.
You then go on to say "every account you can think of is logged in" but you have not said you have re-enabled Jabber or Bonjour.
Can the iMessages account Send anything ?
Start a New Message and choose a contact (I always use my iPhone Number for this) and see if the "To" Spot turns the name or Number red.
If it does then try other people and if they go red iMessages is not logged in.
If it does get past this stage do you get an error message and Exclamation Mar next to the sent item Saying "No Delivered" ?
This also means you are not logged in.
If this is the case Sign Out in Messages of the Apple ID and then restart Messages.
Add back the Apple ID and confirm Password.
Has anything changed ?
Report back at this point.
It may not change anything but will helps us on a route to a fix.
There are several steps that need testing before you might need to contact Apple to reset the server.
9:23 pm Monday; May 2, 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)
