soundproofing apartment floor
As you build up the finished floor higher, you cahnge the dismount onto the top step of the stairs, possibly to something unsafe or against codes. You might end up needing to rebuild the stairs too then.
Spray urethene foam from under will help the sound transmissions.
Sound travels more through solids so any separations will help prevent transfer. We use strapping under the joists (1x4's @16" OC perpendicular to joists) to isolate the drywall ceiling from the solids of the joists.
Mass will absorb vibrations so a double thickness of drywall will not only give better fire protection, but will prevent some sound transmission.
Homasote makes a good isolation/absorbtion ply for between flooring materials. Carpet will cushion footfalls.