I have done extensive searching on Paizo forums, enworld, d20srd, and dorkistan... and I am still confused. So here is my scenario and questions:
A lion animal companion grapples opponent. At the beginning of its next turn, it maintains its grapple as a standard action, then as a part of that action the lion may damage (without an attack roll) the opponent with a natural attack (let's say bite), then use its special attack Rake as normal, attack rolls and damage rolls plus respective bonuses.
So my question is, does the lion's turn look like maintain/bite damage/claw attack/claw attack?
Now let's replace the bite damage with the 2 claws damage. Would the turn look like: maintain/claw damage/claw damage/claw attack/claw attack?
I am unable to find anything that says anyone can normally make an attack while grappling, the only thing I see is that you can inflict damage as part of the standard action of maintaining the grapple... but then, it is a 500+ page book and I am brand spankin' new.
Grapple takes a standard action to maintain. You only get one standard action per round (your choices are one full-round action OR one standard + one move actions, and you can trade the standard for a second move but not the other way around). Which means if your lion (or anything else) maintains its grapple, that's it, that is the only attack it gets that round (more accurately, that's the only standard action it gets).
You are correct, if the grapple is maintained, the lion (or anything else) can apply damage from any one of its natural attacks. That's it. I don't know how you could "replace the bite damage with the 2 claws damage" since you can only do the damage of one of your natural attacks, not two.
Now, the Rake ability allows any creature that has it, including lions, to get two free attacks whenever it successfully maintains a grapple. These are "in addition to" the normal options for grapplers, so it can still bite and make its two "rake" claw attacks.
So your first example was the only correct one: Maintain, Bite damage (no attack roll required, or substitute damage from one claw but bite is better), and Rake (two claw attacks that do not automatically hit).
Just to make sure you have all the resources available, have you checked out the grapple flow chart on the PFSRD website? Link