To take your questions out of order:
Question 2, there is no federal traffic code -- at all. Traffic regulations are strictly a state-level matter.
What there is are the Uniform Vehicle Code (for traffic laws) and the Manual of Uniform Traffic Control Devices (for road signs and markings). These are privately-developed model regulations that most states have adopted to a large degree, either by reference or by incorporating their wording into law.
Question 1, I can find no evidence that North Carolina has adopted any part of the UVC by reference rather than by explicit incorporation. On the other hand, §136-30 adopts the entire MUTCD by reference.
Question 3, the state can delegate rule-making authority to cities, counties, or other entities. North Carolina has done so in parts of their traffic code.
And in response to your question in the comments, the meaning of a solid red arrow is unclear. Section 4D.04(C)(2) of the MUTCD states the following:
Vehicular traffic facing a steady RED ARROW signal indication shall not enter the intersection to make the movement indicated by the arrow and, unless entering the intersection to make another movement permitted by another signal indication, shall stop at a clearly marked stop line; but if there is no stop line, before entering the crosswalk on the near side of the intersection; or if there is no crosswalk, then before entering the intersection; and shall remain stopped until a signal indication or other traffic control device permitting the movement indicated by such RED ARROW is displayed.
This would seem to prohibit a U-turn on a left-pointing red arrow (a U-turn to the left necessarily involves leftward movement), except that §136-30 (the law that adopts the MUTCD) isn't about driving. It's about specifying which signs and signals municipalities and the state Department of Transportation should use.
The section about what drivers should do at lights, §20-158, makes no reference to arrow-shaped steady traffic lights, only circular ones. I think that a driver is required to follow the MUTCD on this by the following chain of laws:
§20-158(b)(5) describing what a driver should do upon encountering an authorized traffic-control device:
When a stop sign, traffic signal, flashing light, or other traffic-control device authorized by subsection (a) of this section requires a vehicle to stop at an intersection, the driver shall stop [at an appropriate spot]
§20-158(a) in turn authorizes the DoT to install traffic signals:
The Department of Transportation, with reference to State highways, and local authorities, with reference to highways under their jurisdiction, are hereby authorized to control vehicles [...] At intersections and other appropriate places, by erecting or installing steady-beam traffic signals and other traffic control devices, signs, or signals.
which leads back to §136-30 specifying that the traffic signals used are those found in the MUTCD.