According to the 2005 Nation Electric Code, can I attach a 3-way junction box to the flooring of my attic?

All electrical connections MUST be in approved connection boxes. These are installed so they are accessible. This allows the box to be installed so it can be accessed WITHOUT destroying the structural integrity of the building. All of the wiring must be protected from physical damage which usually means in walls or ceilings where they can not be damaged with direct blows from any material or tools. In general this means that the junction box can be installed in the attic and protected from damage with good construction practice. The NEC is a minimum standard and the local AHJ may have more stringent codes. Contact a local qualified professional electrician to do the work if you are unsure.

OK, waiting for somebody with current 'Code knowledge to weigh in, 'my amateur take on this. (And I wish I knew what you meant by "three way function", whatever.) Some of it has to do with access to that space. If there's a stairway up there, it's readily accessible for anyone, including kids, vs. a more of a "crawl space", where you like have to get a ladder and go through an opening in a closet ceiling.

Once up there, having a "catwalk" plywood run down the middle is very useful - beats trying to walk carefully across the tops of ceiling joists. I would never put a junction box or naked Romex on that path, but rather be off to a nearby joist. 'Don't want to put the stuff where it gets stepped on, or tripped over, or has storage boxes parked on top of it. I always run with the joists, nicely stapled, proper smooth bends, and drilling holes through this framing rather than running over the top (where it could be stepped on or tripped over). For a main junction box I want to find quickly later, I might use a big one screwed to the top edge of a joist, where it's not buried in insulation; all steel, no plastic. And, if not there already, I add a light or two & switch close to the entry point to the space. (Nice to see without handling a flashlight.)

The box is not a problem, but "the cable shall be protected by substantial guard strips that are at least as high as the cable".

I no longer have a 2005 code book, being superseded by the 2008, 2011, and 2014 versions, I wouldn't be surprised if the website is out of date, and the city is using the current code. Either way I don't think there has been substantial changes to sections 300 wiring methods, 314 boxes, 320 AC Cable, or 334 Non-Metalic Cable.

The current 2014 NEC section on junction boxes doesn't mention attics, but Section 334 that deals with the NM cable attached to the box says that attic installation has to comply with section 320.23 that deals with Type AC cable.

334.23 In Accessible Attics. The installation of cable in accessible attics or roof spaces shall also comply with 320.23.

320.23 In Accessible Attics. Type AC cables in accessible attics or roof spaces shall be installed as specified in 320.23(A) and (B).

(A) Where Run Across the Top of Floor Joists. Where run across the top of floor joists, or within 2.1 m (7 ft) of floor or floor joists across the face of rafters or studding, in attics and roof spaces that are accessible, the ***cable shall be protected by substantial guard strips that are at least as high as the cable.*** Where this space is not accessible by permanent stairs or ladders, protection shall only be required within 1.8 m (6 ft) of the nearest edge of the scuttle hole or attic entrance.

(B) Cable Installed Parallel to Framing Members. Where the cable is installed parallel to the sides of rafters, studs, or floor joists, neither guard strips nor running boards shall be required, and the installation shall also comply with 300.4(D).

300.4(D) Cables and Raceways Parallel to Framing Members and Furring Strips. In both exposed and concealed locations, where a cable or raceway-type wiring method is installed parallel to framing members, such as joists, rafters, or studs, or is installed parallel to furring strips, the cable or raceway shall be installed and supported so that the nearest outside surface of the cable or raceway is not less than 32 mm (1 1/4 in.) from the nearest edge of the framing member or furring strips where nails or screws are likely to penetrate. Where this distance cannot be maintained, the cable or raceway shall be protected from penetration by nails or screws by a steel plate, sleeve, or equivalent at least 1.6 mm (1/16 in.) thick.

So you may be able to, but realistically you would bee better off attaching to the sides of the studs if you can. And as Jim mentioned the local Authority Having Jurisdiction may have established additional rules or may have a very strict interpretation that seems hard to justify based on your or my reading of the rules.

if you are using conduit, then you can put the junction box on the plywood. If, however, you are using romex wire, then the junction box has to be below the plywood. Use a cats paw ( http://i.ebayimg.com/00/s/NTU2WDc3NQ==/z/1dUAAOSwz... ) to remove the nails of the plywood to open it up to run your wiring under it. Mount the j-box on one of the ceiling joists below the floor line. Do not connect j-boxes to rafters overhead. but you can use the rafter support braces as an anchoring point for the j-boxes. The romex should not be running on top of the plywood.

Romex is approved to only be run in walls and attics, but not exposed to where they can be mechanically damaged!