| Noesanity |
1) 100% yes.
2) yes
3) yes [as normal]
4) The feat is applied to any applicable natural attacks the new form has. If the Shifter's Claws normal damage [as listed in the feature] is higher than the improved damage, they may swap it. The feat stops working on Shifter's Claws when they change out of its natural form as that attack no longer exists but is applied to the new form as the feat isn't tied to a particular natural weapon but a natural weapon type. For instance, if you take Improved Natural Attack (claw) and use Mutated Shape to grow a third claw, it instantly gets the damage boost as it's boosting any claws you have but can't do anything for your natural form's claws as you do not have them anymore. In much the same way, changing your size in natural form has no affect wild shaped as it's a new form so it doesn't modify your shifter's claws either but instead either does nothing or is the size change persists, it increased the new forms attacks
I think the problem might be your interpretation of the wording for Improved Natural Strike.
Improved Unarmed Strike, says "The damage for this natural attack increases by one step... as if the creature’s size had increased by one category"
you don't magically get bigger claws or hands, you are just so skilled at using claw attacks that you do damage with claws as if you were a larger creature. This would mean that any form you turn into, be if your base form, tiger, wolf anything with claws will benefit from improved claws, because they are all the same natural attack.
Now, for the primary point of this tread, would improved natural attack work with shifter claws? That is up to your DM, but as the default term for "Shifters Claws" used in the ability description is "Claws" RAW would indicate yes. So unless you were using a different aspect based natural attack, such as hoof or wing attacks, improved claws should work with your claws.
That being said, moving onto the "use of claws in major form." If the creature already has claw attacks, those would also be improved, as previously stated, however if your shifters claws are still better, you would apply the "base" damage to your major forms claws damage... and then modify it with improved natural attack like you would when modifying the major forms base claws. This is also supported by the fact that if there is no "claw" attack in the major form, you can improve 2 natural attacks to the "same damage" as your shifter claws.
so in summary. TL;DR
Improved Natural Attack does work with Shifter claws.
Improved Natural Attack does work with all Wild Shapes, that have claws.
When using wild shape you select the claw damage that is highest from the base form of shifters claws... and then apply Improved Natural Attack as normal.
But at the end of the day, if you are DMing feel free to choose what ever you want. If your DM tells you this is how or is not how any of this works, they are right. This is only an examination of RAW and RAI.