HI,
I am only going to add some notes at this stage.
Some of the info seems at odds with the way Messages works for both iMessages and getting SMS on the the Mac.
I am not disputing what you are saying - I am saying what is happening is outside of my experience with iChat and Messages over the past 12 years.
I know this bit is just semantics but I like to state it any way.
In System Preferences > iCloud there are no settings for iTunes, FaceTime, iBooks, App Store, Messages, Game Center and some other places you can use an Apple ID.
Whilst the Apple ID used in iMessage has to be an iCloud valid one it does NOT have to be the one used in the iCloud settings.
I do appreciate that you have reset the Apple ID in iMessages on the Mac and reset Text Forwarding by the sound of things on the iPhone.
The Process to set up based on the order in which the functions arrive on the devices/OSes
Turn On Messages on the iPhone.
Set the Send as SMS option to include SMS messages via the Carrier.
On the Mac version add the Apple ID (and password).
Test between the two (i.e. the Mac can only send from the Apple ID and the iPhone from the Number)
Add the Apple ID to the iPhone in Messages and make sure both are Send and Receive options.
Enable the WiFi on the iPhone to the same LAN the computer is on.
Enable Text Forwarding.
Select the Mac and that should produce the Code number on the Mac to enter back on the iPhone to link them for SMS
This is working like System Preferences > Sharing > Internet Sharing where the computer gets Internet on one service (Ethernet say) and shares it via another (say Firewire).
IN this case the iPhone is Sharing SMS from it's carrier service to the Mac via WiFi at least from phone to LAN
If the Mac is WiFi and Ethernet connected try it with just one.
If it is WiFi then possibly it could drop out so I would try to exclude that by using Ethernet if possible.
iMessages are held on the server only until the device accepts them.
In the case of multiple devices it is flagged to be sent to each of them.
Once one has accepted it cannot access the iMessage again.
It can be easier to think of them as "Copies"
An iMessages sent to the iPhone Number will go there "first"as it were.
The Sync link caused by adding the Apple ID to the iPhone send a Sync "copy" to the Mac.
The reverse is true of the iMessages addressed to the Apple ID.
Which they are addresses to make a difference as to what the Contact can see regarding the Delivered and Read receipts if you have them set. (i.e. then may not see you have read it if the Mac is Off line and you have seen the "Copy" on the iPhone).
IF...
The Apple ID is not active as a Send and Receive option on the iPhone then this will not work properly.
Same on a Mac.
If for some reason the iPhone Number becomes Unticked in the Send and Receive section on the iPhone then the link for Sync can be broken. You get a real mixed state during this stage until the iPhone Number become de-registered.
There is no clue on the Mac version whether you are logged in or not to the iMessages servers fully or partially.
Only the "Not Registered with iMessage" for all contacts when it fails or Not Delivered feedback text on all iMessages sent.
If the link has failed SMS may still work as that is SMS sharing if looking for a name.
That leaves me mystified about this bit
That incoming text does not show up on my computer. It makes no difference if the incoming text is an SMS or an iMessage ... it doesn't show up on my computer.
As far as I can tell it should not happen that way.
There is obviously some quirk or set up that I have not come across yet.
SMS may sync sometime later depending on what your carrier does about messages being sent.
IT would thing that if the iPhone got them then you cannot get them again - i.e. there is no way to sync these.
If the iMessages server thinks a devices was de-registered for a time then those will not sycn either. They will have been deleted from the server when sent to all registered devices at that time.
10:36 pm Sunday; October 30, 2016
iMac 2.5Ghz i5 2011 (Sierra)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone 6 iOS 10.x and an iPad (2)