No, Code says nowhere that a receptacle-type AFCI or GFCI is required; breaker-type GFCIs and AFCIs are also usable whenever AFCI or GFCI protection is called for by the NEC or otherwise desirable. In fact, circuit breaker manufacturers now offer a device that's both a GFCI and a CAFCI in the same package -- look for a DFCI (Dual-Function Circuit Interruptor) breaker.
As to your clarified question (cites from the 2014 NEC):
The refrigerator receptacle and the dedicated microwave receptacle do not require GFCI protection, as those receptacles are not installed to serve the kitchen countertop surfaces, and likely are more than 6' from the edge of your kitchen sink as well.
The DW circuit, however, does require a GFCI, even though it is hardwired, as per 210.8(D):
(D) Kitchen Dishwasher Branch Circuit. GFCI protection shall be
provided for outlets that supply dishwashers installed in dwelling unit
locations.
Note the word "outlets" in this passage, vs. the use of the term "receptacles" in 210.8(A) -- in Code parlance, an "outlet" is a place where power is tapped from a circuit to serve a utilization device, whether it be a hardwired device, a luminaire, or a cord-and-plug connected device, while a "receptacle" is what you plug cords into.
The disposal does not require GFCI protection either, unless its receptacle is within six feet of the top inside edge of the sink bowl "as the cord flies". (In other words, the receptacle, while not covered by 210.8(A) point 6, may fall under 210.8(A) point 7.)
You have two questions. To the first question of how to get the wire out of the existing box, you would mount a surface-mount box on top of the existing box and put a blank face place on it. Then the plastic wire mold could come out of the raised, blank box and run to the new raised box. The wire must always travel inside an approved box or conduit. You can't just have wire coming out of a hole as you have described.
For the number of wires that you need to run, the current National Electric Code (NEC) guidelines say that all switch boxes need to contain a neutral. If the existing switch box contains a neutral I would use a "4-core cable" which is sold as "14/3 w/ground" (14 gauge, three conductors) or "12/3 w/ground" (for 12 gauge - not common for lighting) to run to the new switch. If the existing box doesn't have a neutral, I'm not sure what code and local rules would say. I would think with such a minor change, you could just extend the wires as they currently are, and you are right that the white wire would need to be marked with black tape for the switch loop.
However, even if that is the "obvious thing to do" or "what most people would do", it might not be legal or proper, so you'll need to get clarification on that. If you have a neutral, extend it to the new switch and you'll be ok.
Best Answer
From the 2014 NEC:
The reference in #2 is a switched receptacle for lighting in a room. If you had a receptacle installed next to the switch and is less than 5 ½ feet above the floor it could be considered serving the floor line. However, practically speaking it would be of limited use to cord connected equipment at the floor line.
So, #3 means you can't use receptacles inside cabinetry to satisfy floor line receptacle spacing. The receptacle inside the home theater stage doesn't count as a floor line receptacle. There is no exception for accessibility within a cabinet through a "hatch", the code just says they can't be used to satisfy the required outlets, period.
Finally, receptacles are required within 6 feet of a door on both sides to prevent people from running extension cords across the doorway. Hopefully, you can see that would be a unsafe practice.
Good luck on your project!