Does the width of the violin neck affect the sound the violin makes?
Technical answer: It probably does, pretty much everything does a little bit.
Practical answer: No. There’s no way any human has hearing sensitive enough to tell the difference.
Is there a prescribed size for necks, or is it more about general smoothness and sturdiness?
It’s all about the smoothness and sturdiness. We do have numbers to work with, but you’ve got room to fudge so long as it feels good in the violinist’s hands and isn’t going to bend or break when you string up the instrument. One of the most common ways a neck will vary is in how rounded it is. Some have cross-sections that are practically a half-circle, while on others it’s more of a very, very rounded triangle.
And following that line of thought, could you make a violin with a deliberate sound by making the neck thicker, or using a heavier or denser wood?
I doubt it. The surface area of the neck is tiny, and what vibrations there are are being dampened by the hand anyway. It would be really hard to determine even if you did too, because no two violins are going to be identical and there’s countless other factors that have a much bigger effect on the sound and can’t be controlled for.
No, if you were going to choose a different size, shape, or wood type, it would be for structural, ergonomic, and aesthetic reasons.