National Electrical Code 2008

300.22 Wiring in Ducts, Plenums, and Other Air-Handling Spaces.
(B) Ducts or Plenums Used for Environmental Air. Only wiring methods consisting of Type MI cable, Type MC cable employing a smooth or corrugated impervious metal sheath without an overall nonmetallic covering, electrical metallic tubing, flexible metallic tubing, intermediate metal conduit, or rigid metal conduit without an overall nonmetallic covering shall be installed in ducts or plenums specifically fabricated to transport environmental air. Flexible metal conduit shall be permitted, in lengths not to exceed 1.2 m (4 ft), to connect physically adjustable equipment and devices permitted to be in these ducts and plenum chambers. The connectors used with flexible metal conduit shall effectively close any openings in the connection. Equipment and devices shall be permitted within such ducts or plenum chambers only if necessary for their direct action upon, or sensing of, the contained air. Where equipment or devices are installed and illumination is necessary to facilitate maintenance and repair, enclosed gasketed-type luminaires shall be permitted.

Which means, NO. The only wiring allowed in ducts, is wiring that is "necessary for their direct action upon, or sensing of, the contained air.".

Mike Holt explains it well in this YouTube video.

If you read article 725, you'll see that it references back to section 300.22.

If you want to go completely amateur, you could just stuff it all in there. However, that's going to result in a significant decrease in airflow and will result in a lot of strain on your air handler's blower motor.

The cross-sectional area of at 16" ID duct is about 200 square inches. That's the number you need to match or exceed to maintain the same possibility of airflow.

Since you are using a rectangular vent and you only have 14 inches of depth, let's figure out how big of a rectangular duct you need. 200 / 14 = 14.3 inches. 14x14? That's pretty close to some of the standard sizes, like 10x20. But since you have to make 2 90 degree bends, up-sizing is a good idea. Also, a larger grate can help reduce airflow noise. So I would go for 12x24, which is almost 50% larger.

So at your local home improvement center, you can try and get a 12x24 filter grate, two 12x24 90 degree rigid metal ducts, and a 12x24 to 16 ID adapter.

However, it's likely you will not find those exact pieces. So what you can do instead is get a few sheets of duct board and make a duct. Duct board is like a rigid piece of insulation rated for HVAC airflow. You will also need a 16 inch collar to attach your flex duct to your new box.

In your case, I would make your duct box extend down through the floor (where you have your two question marks) and attach your flex tube to the face of a duct board. This way you minimize the number of bends you put in the flex tube.