Determine whether Logic Board has actually been replaced I recently had the Logic Board in my Retina MBP 15″ 2012 replaced under a warranty extension program and Apple paid for it. I've reinstalled OS X several times and feel things aren't running properly. Also, I also couldn't find a way to merge my Time Machine old backups with new one.

The problem I have is confirming that I actually got a new board installed. I have not found a way to confirm this. My Mac help isn't helpful and Google is even less helpful. Why is it so hard to confirm that new hardware is really installed after this repair?
You can compare the serial number on the bottom of the laptop casing to the serial number shown in System Report by going to Apple Menu -> About This Mac. When I worked at Apple, it was known that replacement logic boards would carry the new SN (naturally) and that cases from the old SN would link to the new one. 

The serial on the laptop itself is the original SN it shipped with (unless the bottom case has also been replaced) and can be found in very small print near the display hinge. On the Mac, check for an SN that diverges or is not present, as that indicates repair.