NEC 6/12 rule clarification
From the 2014 NEC:

210.52 Dwelling Unit Receptacle Outlets. This section provides requirements for 125-volt, 15- and 20-ampere re-ceptacle outlets. The receptacles required by this section shall be in addition to any receptacle that is:
(1) Part of a luminaire or appliance, or
(2) Controlled by a wall switch in accordance with 210.70(A)(1), Exception No. 1, or
(3) Located within cabinets or cupboards, or
(4) Located more than 1.7 m (5 1⁄2 ft) above the floor

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 &quot;hatch&quot;, the code just says they can't be used to satisfy the required outlets, period.

210.52(A) General Provisions. In every kitchen, family room, dining room, living room, parlor, library, den, sunroom, bedroom, recreation room, or similar room or area of dwelling units, receptacle outlets shall be installed in accordance with the general provisions specified in 210.52(A)(1) through (A)(3).
(1) Spacing. Receptacles shall be installed such that no point measured horizontally along the floor line of any wall space is more than 1.8 m (6 ft) from a receptacle outlet.
(2) Wall Space. As used in this section, a wall space shall include the following:
(1) Any space 600 mm (2 ft) or more in width (including space measured around corners) and unbroken along the floor line by doorways and similar openings, fire-places, and fixed cabinets
(2) The space occupied by fixed panels in exterior walls, excluding sliding panels
(3) The space afforded by fixed room dividers, such as freestanding bar-type counters or railings
(3) Floor Receptacles. Receptacle outlets in floors shall not be counted as part of the required number of receptacle outlets unless located within 450 mm (18 in.) of the wall.

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!