MBP will not register with iCloud servers (and most iCloud features + Handoff and Continuity do not work)

Question

Hi all,
I have an extremely strange issue on my rMBP which has not been solved yet despite over 2 hours on the phone with Apple support, and another hour after being put through to a Senior Advisor, and several online chats.
I have a Apple Certified Refurbished Late 2013 Retina Macbook Pro, which has a 2.3Ghz i7, 16GB of RAM, 500 Flash Storage and the NVIDIA discrete graphics. It was purchased in December 2014 on the Apple Store. It is running 10.10.2.
For some reason, my computer WILL NOT register with iCloud. Messages and Facetime both would not work, my iPhone 4S which is capable of text message forwarding would not recognise it and allow me to turn it on. Handoff, Continuity and Instant Hotspot - basically any of the features to do with iCloud or connecting to another device - do not work.
Originally I believed it was just an issue with Messages and Facetime so I reinstalled the OS back to 10.10.1, but that did not solve the problem. All my Messages contacts kept on coming back as 'not registered with iMessage' and I could not turn on SMS Relay either.
I contacted Apple on Wednesday night and ended up talking to several different people, the Senior Advisor was the most helpful - really tried to help me figure out what was wrong, and to fix it - wonderful help, really was. They used screen sharing and were quite confused. They figured out as well that Handoff, Instant Hotspot, and all those other features don't work either. At this point, they made me update to 10.10.2 to see if it was an issue with the update, but to no avail.
Eventually, the SA checked my iCloud account and saw that the computer hadn't even connected to iCloud, and wasn't even showing up as one of my devices, despite the fact that I had signed into iCloud on this computer.
The SA told me that he suspected it must have been something to do with hardware, but didn't know what it could be. He said the only time he had seen this before was on a refurbished laptop when the serial number hadn't been put back on, but my serial number was there, so he said it must have been something to do with Bluetooth or Wifi.
Wireless seems to work completely fine, but there is a strange thing with Bluetooth in which within System report says that the computer is not discoverable or connectable, despite the fact that in the Menu Bar it says it is on and discoverable. It becomes discoverable and connectable once you toggle the switch in System Preferences.
I was planning to book a Genius Bar Appointment as suggested by the SA, but something strange happened today - all of a sudden, some (not all of them) of my iMessages from earlier today came through. Still no working Handoff or Facetime or Continuity, but now I am starting to wonder what is really wrong.
Despite the fact they said it was hardware (especially Bluetooth), what could explain why it (seldom) or will not register with iCloud? Can anybody shed some light on the situation and whether a Genius Bar appointment could help fix it?
Cheers and thanks in advance.

Answer 1

Needed to add some clarification here:
iCloud shows me as signed in, but Calendars are NOT checked
Internet Accounts shows Calendars as syncing