Hi,
I have my Mac version set to "Start New Conversations From" the Apple ID and the iPhone is set to do the same but from it's number.
I don't have an issue with Sync running this way round.
iMessages are not effected by the setting in Messages Menu > Preferences > General section that says "When I Quit, set the Status to Off Line".
This only effects Bonjour, AIM, Jabber (including Google) and Yahoo accounts - the Ones with Buddy Lists that actually have Status.
Setting this can mean that those accounts do not show Off Line notifications as the process that "listens" for them is not running.
Other than turning the Mac Off the iMessages part cannot be stopped.
Things that can delay the Start up can be lots of files and icons on the Desktop. These get treated the same way opening windows that were open at Shut Down.
The more Apps that have to be opened as well can effect how long it is before items like the iMessages Notifications can get triggered.
The Start Up time can vary from computer to computer depending what is installed and what is allowed to "call Home" during the Start up process.
Running Little Snitch can tell what the computer is trying to connect to and what incoming connections are also being made.
However during the Start up process not dealing with the notifications of Little Snitch can be it's own hold up.
You will notice though that the process for the iMessage Login is quite late in the whole set of things that the computer does during start up.

8:57 pm Tuesday; July 26, 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)