How to remove rusted remains of kitchen sink tailpiece?
I think I would just be inclined to replace both the tail piece and the PVC pipe (and trap if necessary).  You might be able to get it out without damaging the pipe but more then likely you will at minimum, rough up the surface inside which might just attract gunk to stick to it over time. It's probably easier than trying to remove it too.  This should not be an expensive repair.

As far as what to replace it with, you need to measure the diameter of the existing piece and the distance between the trap and the sink. Make sure to account for the overlap needed for the fittings - better to have it a bit too long and cut it down.