In the rake rules it specifies that in order to get the extra claw attacks, the creature must begin the round already grappling its opponent. Does it has to be the one controlling the grapple in order to use this ability?
For instance, if I grapple a lion, can it rake me on its turn without winning a grapple check?
Yea, same as how you can still make a full attack with a light weapon and get your iterative attacks on someone that's got you grappled, something with rake gets those bonus attacks.