Rapier 1d8 (Finesse weapons, light)
. . .
Also, the rapier should be a light weapon. You should be able to hold a dagger in your offhand.
I don't understand what you're getting at with this. Rapiers aren't light weapons.
Or are you saying that light weapons should do 1d8 damage (instead of the 1d6 that shortswords and scimitars do)?
As for the dagger thing, I agree. But rather than nerfing the rapier down into "just another shortsword," my fix would be to give the dagger itself a special rule that would let it be paired with any other weapon.