Ganondorf is unaware of the cycle except in Wind Waker where it is hinted at in Hyrule Historia that he wishes to put an end to it by assembling the Triforce.
In Twilight Princess he is unaware of any cycle and he is reborn in FSA and would be unaware of his past in TP. After that we don't know...
Yeah DarkestLink said it, anyone alive today is from a bloodline that has lasted for thousands of years. That's why it's not so strange for the royal bloodline of Hyrule to still exist even ten millennia after Skyward Sword. I will say it is unnusual for the royal family to still be in power so...