The poly simply needs to cure for a while. Even after it "dries" and is safe to work on, the solvents that keep the polymers in suspension aren't all gone; they'll continue to evaporate, and the clear coat will fully harden in time. Check the can for your particular product; it should have a full cure time. It could take up to a month if you laid on a really thick coat and it's humid in your area.
Contrary to intuition, you can wax furniture that has a poly coat, much like you can wax a car that has a clearcoat. Just pick up a can of Minwax, apply and buff, then let dry overnight. This additional protective coating will be temporary, but hard, water-repellent and slick.
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.
Best Answer
From what you've told us, it sounds like you are trying a number of good ideas to help mitigate the smell and lingering off-gassing but maybe not optimally.
You actually want to put the space heaters away, in fact, many apartments forbid their use. As long as they provide functioning heat, they can prohibit their use. If it isn't prohibited then use it, but realize they are a safety hazard in more than one way, especially if your fumes are as potent as you suggest.
Just close the windows, get the heat cranked up a little higher, and get a fan to circulate the air. Put water sources away, dry the sink, etc. Don't add unnecessary water to the air. Locate the strongest source of the odor and get the fan circulating air in that room.
Vacate the premises for a few hours, then when you get home, go ahead open two windows, preferrably opposite sides of the space, and put the fan blowing the air directly out for 10-15 minutes. Close the windows, repeat.
I'm not suggesting you do this as an option, but I want you to understand an example of how long a smell can last despite the best of intentions & I only came to this after a few years of frustration:
You indicate that you know you can take legal action, but honestly, I'm not so sure that is true. Every area has laws and regulations to follow, but (even if this was subsidized housing), it doesn't sound like the landlord failed to provide livable living space.
If the smell is indeed paint/poly/etc., the instructions&use of most of these products is typically in the span of 1-2 days, so weeks later, its supposed to be safe even if stinky. You can seek out staying at a friends or families, but I wouldn't expect any compensation or recourse from the landlord unless they did something wrong, and its not advisable to throw accusations unless you have a basis.
Keep doing what you are doing, and see if you can pinpoint the issue. Warm air & ventilation will help. If you can locate the source and it isn't paint/poly, then discuss with the landlord. If they are renovating other nearby apartments then request them to wait holdoff spring to use chemicals if at all possible.