I've started Skyrim from the beginning - last time I completed the golden claw mission and gave it back....is this a mistake? surely you have to fulfil the mission by giving the golden claw back? Do you have to keep all the subsequent claws?

Use them once and then you can get rid of them.

I collect them, though, and display them on my house. I stole the Golden Claw back as soon as I returned it.

I keep all the claws for the hell of it. Heck, anything that has a unique name in Skyrim that I can keep I put in my chest.

Nope, they become irrelevant once you unlock the associated door.

Though one time I went back into Bleak Falls Barrow and the claw door was locked and shut again...