MBP: F1-F9 & Right Shift Not Responding
Okay, it's a bit hard to try and troubleshoot this via lots of comments to and fro, so I will use this to try and get you sorted.

I'm assuming that since you've already tried formatting the disk and reinstalling OS X that you already have a recent backup. If not, do one now.

Try an external keyboard

As I mentioned in the comments, have you tried using an external keyboard? Any USB keyboard will suffice - give it a go and report back your findings. Basically, troubleshooting is a process of elimination. This simple test will greatly reduce the number of possibilities and help troubleshoot your issue. 

Reset your PRAM/NVRAM

Older Macs had what's called Parameter RAM (PRAM), newer Macs use Non-Volatile Random-Access Memory (NVRAM). Here’s how to reset your PRAM/NVRAM:


Shut down your machine. Yes, a full shut down, not just logging out.
Press the power button and then press the command-option-p-r keys. You have to make sure you press these keys before the gray screen appears or it won’t work.
Hold those keys down until your Mac reboots again and you here the startup chime.
Let go of the keys and let your Mac reboot normally.


Note: When you log back in you may need to readjust some of your system preferences (e.g. mouse speed, time and date/timezone, etc).

Reset the SMC

Your computer's System Management Controller (SMC) can also be reset. The instructions for resetting your SMC depend on three things:


Does your device run from a built-in battery? (In your case, yes)
Does your device run from a removable battery? 
Does your device just run off power from the wall?


Now, for the sake of others who may come here looking for answers to the same issue, I will provide the process for all three. 

For MacBook Pros, MacBook Airs, and MacBooks where you can’t remove the battery on your own (i.e. it's a built-in battery), here’s what you do:


Shut down your computer
Keep the MagSafe adapter (power cable) plugged in.
Press at the same time shift-option-control (on the left side of the keyboard) and the power button
Let go
Turn your computer back on with the power button.


For MacBook Pros, MacBooks etc that you can remove the battery from, here’s what you do:


Shut down your computer
Disconnect the MagSafe plug (power cable) from the computer
Remove the battery
Press the power button for 5 seconds and release
Put the battery back in
Reconnect the MagSafe cord (or power cable)
Turn your computer back on with the power button


For iMacs, Mac Pros, Mac Minis, etc that only run from a power point in the wall


Shut down your computer
Unplug it from the power
Press and hold the power button for 5 seconds
Plug it back in and turn it on.


Hope this helps. Please report back and let us know.