How to fix the humidity problem in the home

Your best option is sheet vinyl. Next to that would be tile or concrete.

Most other flooring has seams which will eventually leak. Yes, you can place vinyl tiles, or even laminate flooring, if you appropriately glue/seal all the seams. Eventually the seams will leak, and in the case of laminate surface scratches must be sealed immediately to avoid damage to the laminate below the waterproof surface.

A single sheet, if possible, will provide continuous leak protection across the entire floor surface. If it's too wide for a single sheet, then a single seam is easier to keep sealed than the multiple seams found in other flooring choices.

Tile and concrete are both good options as well. Regular maintenance and upkeep will keep them in good waterproof operation for the life of the house, while vinyl will need to be replaced every decade or two.

Even though it's designed to be an unheated space, you should consider adding a vapor barrier and insulation to the flooring area. This will reduce the likelihood of later rotting or water condensation.

Keep in mind that outdoor carpet requires a sealed or well-draining surface. Your current plywood would have to be treated and sealed for the outdoor carpeting to be useful. The carpeting alone won't prevent damage to the subfloor - it will only avoid damage itself when exposed to water.

In your situation you need to protect both the flooring and the subfloor.

Regarding the cold temperatures, newer vinyl floors are much more flexible and resilient even in the fact of extreme cold weather. You'll need to specify that both the flooring and the adhesive be tolerant to freeze-thaw cycles.

Be certain that your subfloor is substantial and secure. Cracking occurs where stresses increase significantly on one area of the floor. As long as the vinyl is well-adhered to the subfloor, and the subfloor is relatively stable, then the stresses from expansion and contraction will be spread evenly across the entire surface. While these forces are large, they won't overcome the strength of the vinyl. If the subfloor develops a large gap, then the stresses in the vinyl would be greater around that gap and that's where a tear might start.

So your flooring really starts with a good, stable, strong subfloor. What you put on top, then, will be fine as long as it can deal with the water exposure.

This is going to be sort of a comment, or at least not a complete answer.

When designing the system, the goal is not to create a breeze through the space. The goal is to control pressure equalization in such a way that the air is mixed and dehumidified (conditioned).

Having one large return at one end of the space, and a large return at the other may work. But it's not going to be as efficient, and will require maintaining an open area for which the air to flow (no closed doors).

Let's back up a bit, and think about how the system works...

The fan (or blower) is going to create a high pressure area "in front" of it, which causes the pressure in the supply ducts to increase. The higher pressure air flows from the supply ducts, in an attempt to equalize the pressure between the supply ducts and the rooms. At the same time, the fan (blower) has also created a low pressure zone "behind" it. The higher pressure air in the rooms is forced into the return ducts, in an effort to equalize the pressure between the rooms and the return ducts.

So the system works by forcing conditioned air into the rooms, and having unconditioned air pushed into the system.

Wherever you have a supply duct, you want to have and unobstructed return duct. Otherwise you'll end up with a high pressure area in one place, and a low pressure area in another.

In your example, if you have a single supply duct feeding the kitchen, and a single return in the bedroom. What happens if the bedroom door is closed? In fact, the bedroom door in your drawing would have a tendency to be pushed closed, if you installed the system the way you've described. You'll be creating a high pressure zone in the kitchen and living room, and a low pressure area in the bedroom. Which means the door could be closed by the pressure differences.

You'll want to install a supply and return in each separate area. In your case that would mean installing a supply and return in the living room/kitchen, bedroom, and storage area.

Unfortunately, duct design is a bit of science, and a bit of art. Covering all the information you'd need to know, would be nearly impossible in this short answer format. In my opinion, duct design is not a job for an amateur. At the very least, you'll want a pro to assist in that aspect of the job.

I know I've rambled a bit, and I'm sure you don't want somebody to tell you not to do it yourself. But in my opinion, it's better to have it done right, then to have it done cheap. Aside from all that I've mentioned. It's impossible for somebody to design a duct system without knowing the equipment you're going to use, or the actual dimensions of the space. So there's no way anybody on this site could provide a duct design for you, with the information you've provided.

My advice... Get an HVAC contractor to design the system for you, then you can do the work of installing it yourself.