Hi, What is the diameter of the mounting plate?

The diameter of the bottom plate with the pin-lock connection for a RAP-B-379BNHU is 1.25".

I recently purchased the large tough-claw with a 1.5" C size ball. However, I needed the 1" B size ball. Can I purchase this B size Pin Lock Ball Adapter and use the same mounting screw that came with the orignial large tough claw and C size ball?

Yes, you can use the RAP-B-397BNHU with the existing screw that keeps the C size version of that pin lock ball adapter in place.  It will require more tightening as the screw with the C size ball is slightly longer, but it will work.

How do I get this to mount to my Tough-Claw? The screw that comes with it doesn't go through the top (it came in through the bottom). What size bolt should I get?

This particular SKU does not come with any mounting screws. The screw that is used in the main assembly is a #10-24 x 1.75" stainless steel machine screw. This can be sourced through any local hardware store and should allow you to connect the ball to any Tough-Claw base.