Will I have problems with smooth gaming when storing the games on the NAS and would the Pi 3+ make a difference in that department?
I'd say the difference you'll see is negligible, since once the game/rom is loaded in the emulator, there's little network activity or file transfers over the network. One exception would be - maybe - CD based games where the game is loaded from the CD images (.iso/.bin/.chd). The network speed would help the loading times. But if you are already content with the Pi 3 and the current setup (ROMs on the NAS), it wouldn't make a noticeable upgrade.