Flooring – What kind of flooring is this

I had a similar situation when I redid my basement - couldn't put baseboard in to hide the gap between the doorway and the laminate flooring at the entry door. The most common solution I've seen in images I found on the web and in my chosen home improvement store was to use color coordinated T-moulding along the doorway right up against the bottom sill. I'd add an image, but I don't have one handy, alas.

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.