Is synchro rev on by default when you start the car?
 

Does it automatically turn to ON each time you start the car? Or does it remember the last setting?

And also, what happens if you try to blip the throttle while it's on?
Does it know that you're blipping and it won't match for you?

The car remembers the last setting...ex: if i have sports mode on and turn the car off; the sports mode will be on when the car is turned on again. This is indicated by the "S" in the tach.
blipping...it will still blip because it matches to the perfect rev.
i personally have tried to "trick" the rev match by putting it in neutral, moving around the shifter...but the computer just holds the rev for a moment then goes down to idle. its pretty smart. fun to have.

in other words, if it is on, it will override your throttle input (up or down) to put the engine revs in the right range for the shift?

3rd (modified by Nissan) law of robotics - A robot must protect its own existence as long as such protection does not conflict with the First or Second Law. This does not include the Synchro-rev match: trickers will be SHOT!

If you have it in S-mode and go to blip your own throttle on a downshift (out of instinct/habit or whatever), the computer will override your input and still blip it to the correct level.