Can you install grouted floor tile over padding?
In almost all cases this is a no.  There are waterproofing systems you can use but not flooring pad.  When you lay your tile the pad will move a little and cause issues.  In essence with a pad you are creating a giant floating tile floor.  

Note: To reduce sound from the floor you want to do two things.


Build up the subfloor.  Adding an inch of plywood plus 1/2 inch durarock will take out a good % of any sound.
Add insulation in the cavity below floor.  


The padding you are thinking about leaving might reduce sound by 1%.  Sound is due to weight disbursement - generally speaking.  A 200 pound person running around will make sound.  10 pounds of padding underneath them will have no affect.  I am sure someone can comment on the science of this.  But the two things you want to do are create a buffer (pad is thin and almost no buffer) and keep your joists from moving (pad does nothing at all there).