This is an acceptable solution for multi-tenant buildings where the exhaust fans from multiple units vent in to a common duct. The booster fan in the floor vent makes sure that the air pressure in the room being vented is higher than the common duct. The booster fan can pull air through the central AC/hear air handler even if it's not running. This lightens the load on the exhaust fan.
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.
Best Answer
Non PT wood will be fine.
I would build a 32x22" frame out of 2x4's but I would put a 3/4" square notch around the inside diameter of the frame and set the filter into that notch. That will give you a nice seal for the filter but prevent it from falling through. If necessary, add one or two cross braces across the inside of the frame cut to the depth of the notch so the filter has extra support.