Sylveon is pretty awesome, but you guys have the wrong idea with Calm Mind / Dual Screens. Those two sets are decent, but they either have some sort of major flaw or they're outclassed by some other Pokemon. I would pretty much always use CM Clefable (who btw is fucking amazing; try it if you haven't already) over CM Sylveon, and I would pretty much always use something like Latias or Latios over Dual Screens Sylveon (and dual screens isnt even that great of a strategy, especially on a slow Pokemon like Sylveon) I mean both sets are decent, but they're not nearly as good as a defensive set or even a Choice Specs set.
The real good set of Sylveon is the standard 252 HP / 4 SpA / 252 SpD with a Calm Nature. Sylveon is a pretty awesome defensive Pokemon; it has very good special bulk with 95 / 130 defenses, reliable recovery as well as Heal Bell, and it actually hits pretty hard back. Sylveon is one of the strongest defensive Pokemon you will find, which means that Sylveon can retaliate vs various defensive Pokemon pretty easily. Zapdos (just using it as an example because of its bulk + neutrality), for example, takes around 50% from uninvested Pixilate Hyper Voice.
As for Choice Specs, I haven't tried it yet, but it seems pretty decent in theory as that Pixilate Hyper Voice hurts. You would use something like Hyper Voice / Shadow Ball / Hidden Power Ground / Baton Pass if you were to use Choice Specs.
Also please don't use Big Root... it's a really bad item that should not see use on anything. There's a reason it's never used in competitive Pokemon: it's outclassed by Leftovers.
Note I'm talking about post Pokebank. Sylveon is not nearly as enticing to use without Pixilate Hyper Voice and Heal Bell imo.