Romex is a specific brand name for NM cable. NM is not rated for and should not be used inside of conduit. When installing wiring in conduit one would use individual conductors rated THNN/THWN. There is a maximum number of conductors that could go into each size of conduit, but once there are over 3 conductors they must be de-rated.
Any holes drilled between floors for pipe/wire/conduit/etc. must be sealed with fire-rated caulking.
What you are trying to do is not a bad idea, but given that NM cable can easily be pulled through a finished interior wall at any point in the future, it might not be worth bothering.