I'm in Pennsylvania, probably not as bad of winters, but I have an intex as well. Cover laying right on the water is fine. The pillow helps to push ice down into the pool, as opposed to forming a solid sheet across the surface. The pillow will likley move a bit too with water displacement, wind, etc. This is all good for ice break up. What will damage your liner is solid ice forming from all sides in toward center - ice expands and it could rip your liner.
Need to drain pool below the intake/output ports on the pool, and disconnect your pump and filter, bring them inside for the winter along with all plumbing - garage or basement.
Prior to draining, make sure that you check your balancing chemicals, clean the pool of all foreign objects, SLAM, and add winterizing chemicals. Give the pool enough time to do a full circulation with the pump running after adding the winterizing chemicals before draining. 2" below your in/out ports on the pool should be far enough for the drain.
cover - I use clips to hold it on to the top bar of the pool, or to support legs, as opposed to trying to weigh-down the cover, which I tried first year in an effort to let any water/snow just run off the sides. It doesn't work. let cover lay flat on water. Another reason you need the air pillow, all the weight of rainwater/snow/ice, etc on top of the cover will push out all of the pool water eventually, up the sides and out the holes by the top-bar. The air pillow provides support/displacement for the weight/pressure of the water/etc on top of the cover. As stated above, very good idea to get a pump for the top of the cover. I don't have an automatic one, mainly because I think if any of them are in water and freeze it will likely break the pump. kind of a pain in the butt however, it is less-work per week usually than when the pool is open, tending to pump backwashing, chlorine/chemicals, etc. What I do is put my pump in when I need to siphon off the top, plug it in for a couple seconds to get the water flowing, unplug it and walk away. if you decide to leave it plugged in to get faster water flow, need to watch it. Just like your pool pump, if that cover pump continues to run without load (water) to transfer it will burn out the motor. Obviously, if the water on top of the pool is frozen you'll have to wait for it to thaw before you can pump it off, which is why it is a good idea to try to stay somewhat up to speed with it.
Another reason you want to have the cover is for the pollen drop in the spring. Don't make the mistake of opening too early, you don't want to have to deal with all of that in your pool. Wait until it is done before opening. Usually 2nd or 3rd week of May for me is good. Best of luck!