How do I reroute my existing bathroom piping for a pedestal sink?
The stud you would use a hole bit that is the same size as your piping to cut a circular hole.  You can pick up these bits at big box or a place like Harbor Freight in the US.  

You do need to make sure that the stud is not load bearing!  If it is or could be we need more info and pictures.

It really looks like you have PVC but we can't see everything.  If you need to go from PVC to cast iron use a coupling.