As Brittany stated earlier, we only use hot-dipped galvanized in this application. It is not a code issue but it does make a huge impact on quality and what will be seen for years to come.

The correct nail to be used is a Hot dipped galvanized nail. Nail diameter and size are different on all applications . A regular galvanized nail is only electo galvanized meaning as soon as the driver from the vail gun strikes the nail that part is no longer galvanized . Always use hot dipped for all aplications and you'll be safe. Sometimes We have to go repair entire houses where siders us the electro galvanized instead of paying the $10 more for the correct nails and you will see the nail heads actually rusting. Great question though. Hoped this help. www.greenbuiltwa.com

Unfortunately, insulation and sound proofing is not required per code between conditioned spaces in this case.

It is also worthy of note that additional insulation between the floor will do very little to impact the noise tranmission as a majority is carried via vibrations on the framing. 

There are applications and techiques that can can be deployed.  All of them will require removal of the existing ceiling if you stay with a drywall finish.  You can run resilient channel in addition to sound de-coupling isolators. 

It is not an easy or cheap bullet fix in this case.