Each natural attack is independent by rules. This is one of those situations where you GM is using a "common sense" ruling to override rules. It, further, is one of those "common sense" rulings that is at odds with how the real world actually works since tigers (and cats in general) do frequently attack with a bite and then use their claws to help hold their prey after the fact.
No, yes, and right, respectively. That rule totally does exist... for the rake attack, which is (probably) supposed to represent the claws on the hind feet of the cat. Tiger rake is only usuable in a grapple or on a pounce.
It is, in the end, the GM's call, but it's not the standard rule in the game. They may be thinking of housecats, who swipe their paw at things before trying to grab them with their mouth, but miss that those cats are playing, not hunting. Hunting they'll just grab and snap the neck with their mouth, no claws necessary.
I think there is a preference for grabbing, but it isn't absolutely necessary. Unlike in game terms, (many predator) claws are designed less for damage and more for acting as tiny hooks to latch onto the flesh of a creature. Cat really emphasize this trait.
The general advantage of claws in that case is that the tiger can basically tackles the deer, and then squeeze its arms and lets the claws latch onto whatever. Little precision required to get the prey where you want it- to the end game. Compare that to a bite, which typically needs to go to the jugular if you want an instant- otherwise, the deer will just run with slightly less butt if it can.
Of course, all this is for real life mechanics. Not for game mechanics which are based off of media presentation, where claws can slice through steel, and bites always take a head off.
On a related note, if you toss a Natural 1 while performing the claw/claw/bite, would a fumble toss out all three attacks? If so, should rolling the attacks separately help to get in one or two attacks before a hypothetical 1 on the 2nd or 3rd attack?
I don't know how the fumble rule here compares to the "must have a claw hit to make the bite attack." [Note: we're not using that]
My guess is they come out about even, unless they are both being applied, in which case you really need some Hero Points for rerolls or some Luck domain effects.
On a related note, if you toss a Natural 1 while performing the claw/claw/bite, would a fumble toss out all three attacks?That was a rule in 2nd Ed AD&D. It's not a Pathfinder rule.
It wasn't even a rule then, just a very common house rule.