Firstly, I can't drop any of my Amulets of Talos (3) because the games says that it is a quest item. What quest would I have to do to get the game to allow me to drop them?

Secondly, is there any way to get the "Time between shouts is reduced by..." effect for enchanting? And where would I find an item to do that?

A quest given by the Jarl of Solitude involves an Amulet of Talos, that could be your problem =/.