1. I'm assuming you bought a circular polarizer and not a linear one.
2. Foggy ... I can only guess, but I'm guessing that the fog was coming from moisture (condensation trapped between the filter and the front element of your lens.) on the filter - going from warm to cold will do that. Did your pictures get clearer after the filter had been mounted for a while?
3. A polarizing filter actually has two rings - the one that screws into the lens and the outer one that rotates. It's tricky, but in time you'll get the hang of rotating without twisting the mounting ring and therefore the focus. (Note: If your filter doesn't have a rotating outer ring, it's a linear and not a circular polarizer. Get your money back and buy the right thing)
I'm sure you already know this, but just in case: the polarizing effect of this filter operates on it's relative angle to the light source (the sun). So every time you change the direction in which the camera is pointed you must re-orient the filter. Otherwise, it's just operating as a darkening filter - thus causing low exposure. A polarizer does affect exposure, reducing your f stop by as much as one full stop, maybe more. Your camera's light meter takes care of this, but unless you're aware of what's happening it's possible to be shooting much lower than you thought.
How to adjust a polarizer? Point in the direction you're going to shoot, then, while looking through the viewfinder, rotate the outer ring until you see a slight darkening of the image. It's very subtle, but you can see it. that the indicator that the filter is properly aligned and most effective.
Hope this helps!