Regarding NYCDOT/NYSDOT, the split is basically thus: NYCDOT handles everything a NYSDOT maintenance residency would handle everywhere else.
Regarding truck routes, I happened to hear our Regional Planning and Program Manager mention in conversation that NYSDOT doesn't normally get involved with truck routes. Given that and the mention of Truck US 6, I also wonder about Truck NY 30, since that section of NY 30 has a legislative ban on trucks.