To the strings cutting deeper into the nut, absolutely. The stock SE nuts are, in my honest opinion, doodoo. I've replaced every single one on any SE I've owned. It is possible that might have been our culprit, but without doing proper truss rod adjustments first and letting her settle to eliminate that as a problem, itll be like the tootsie pop question for all time. The world will never know.
Lowest action on an SE custom without audible string buzz. My 35th has .050 treble side action. Its a hair over 3/64. Personally, I think it can go lower, but there are some problem children on the fretboard that will need a fret eraser. Bear in mind, when I first got it it was 7/64" and still buzzed.
Lowest I ever got is 2/64, or .031. Thats my Pauls Guitar SE. Beyond that, its about as low I think one can go before it goes into sort of unplayable territory, where you will get buzz no matter what.
One more thing, I would suggest that if you do pay for a setup on this new guitar, dont do it for at least a week, maybe two. The reason being is that any setup work should can be spoiled by a guitar that is still acclimating, but the chances go down significantly if that process is basically done.