I am using Maya 2014 Student Edition, so I don't think that the version is the problem.

I did manage to get my claw to pick items up using dynamic parenting constraints in a script. The claw fingers close like I want, but they aren't rigid. Since the parenting works okay, maybe the fingers don't need to be rigid. I don't notice them clipping thorough the picked up object at all.

Now all I need to figure out is how to get gravity to take over once the item in the claw is dropped.

Thank you for offering your advice on how to fix the rigid body issue. While I am still a bit confused about the rigid bodies, I think I have a better understanding than I did before!