Airbus approach capability
 

What is the logic the Airbus uses to move from CAT 1 to CAT 3 DUAL. Is it RA driven?

Partly. Autopilot availability and other redundancy considerations are used to decide landing capability - only one of which is Rad Alt.

Others include Hydraulics, Yaw damper, FBW computers and NWS just to name a few.

There is a whole list of systems required for CAT 2, CAT 3A/B in the QRH.

Not specifically, but usually the Rad Alt criteria is the last of the monitored criteria to be fulfilled as they don't pick up a signal until around 5000' AGL.

Pretty much the same for the modern Boeing birds.