I assume you mean the woods used for both fingerboard and the "core" of the neck, as well as neck reinforcement technologies (e.g. truss rods, carbon fiber, etc.). One thing I have noticed is that vibrations in the neck translate to the tone captured by magnetic pickups. As an example I built a neck where the truss rod was not properly glued in and it rattled during playing, and the rattling was readily apparent in the amplified sound.
Yes. Especially with regards to rigidity. More in the rigidity and density than the actual "type" of materials used, though certainly you can generalize that a graphite neck will influence the sound in a different way than a maple/rosewood neck.
For me personally, as long as it has adequate rigidity and I don't need to make constant adjustments, I'm happy. I worry less about "rosewood vs maple" type debates and more about "am I gonna constantly have to mess with this green wood neck.
PS: rattling is bad.