Hi,
What you describe is the app working as it should, despite the fact you may not like it.
The "sync" as Apple calls it in iMessages is only "Display on all devices".
Anything your do to an iMessage on any other device is immaterial.
Only iMessages specifically addressed to the Apple ID are likely to get marked as Unread.
It works like this :-
An iMessages is sent to you, lets say your iPhone Number.
It is duplicated at the iMessages server for each device that is listed as using the iPhone Number.
Each iMessages is then pushed at the device it is set up for and will continue to do so until that device "answers" and accepts the iMessage.
As each device logs in Uniquely the iMessages can only be sent to those devices in the same unique manner.
A "Copy" fro the Mac cannot be accessed or read by the iPhone as they are not "fetched" like Mail.
Only the addressed ID (In this case the iPhone number) will have the "Read", and "Delivered" alerts seen by the sender.
Other devices will also ignore whether they are new to the device and not show the "Unread" markings.
The delays perceived when the computer is turned On and the Messages app started up remains a mystery as to why they don't all arrive at once.
It is probably some process at the iMessages server end that tends to batch send stored iMessages.
9:38 pm Monday; November 30, 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
