Macbook Pro 11,3 No longer has a serial number
Question 1a:
We don't know all the steps you have taken (and you may not remember all of them very clearly) in your reinstall.  So hard to say why.  The most common causes are some hardware change, but you say this does not apply in your case.  Of the steps you mention using dd is the only unusual step - I would have used Disk Utility in Internet Recovery.
Question 1b:
You have already discovered that some Apple services require a valid serial number. iMessage and various DRM are tied to your serial number and UUID as well as Apple ID.  As an aside, loading a valid serial number is a tricky and important step in getting a Hackintosh to run iMessage.
Question 2:
The best solution is to take the MBP to an Apple Store where they may well fix your problem for free - just say it happened during a macOS upgrade (which is true in your case).
Question 3:
The Blank Board Serializer is an Apple tool and can be used to fix your Serial Number.  It is important to find the version which is compatible with your MBP - I can't help you there.
Have a read of these three links which describe the process which must be following very carefully - the first one has lots of comments from those who have succeeded (or not) using this method.
Roger Sen
What causes Mac to lose its serial number
How to fix missing serial number
Warnings:

However you fix this (either via Apple or yourself) make very sure you enter the correct serial number belonging to your Mac.  You only get one chance to get it right.
If you do it yourself, be prepared that you might get it wrong and brick your MBP.

Additional warning:

Read @DanRan's own solution and heed the warnings about the need to unregister the Mac and unauthorise from any services which have been used with the incorrect or blank serial number.