1. You will be issued the regular boots. No gortex until SQ or MOC training.
2. Bring your own PT clothing. The course sets its own standard. Typically, it is grey track pants / shorts and an OD t-shirt. You may be able to get away with black shorts. It all depends on your course staff. I've been on a course were they didn't care at all. But, if you get the urge to buy PT clothing before you are actually on course, than you can't go wrong with grey track pants, grey shorts, and an OD t-shirt.
3. Yes, you will run in the snow, the slush, etc. And, you will do push-ups in it as well. Some days, you may even grace a gym with your recruit presence -- it all depends on whether your course staff are a bunch of sadistic bastards or not, and whether facilities are available. You'll just have to learn to suck it up and take whatever they throw at you.
4. You don't need a sweater to train outside. Wear your fleece top, with the poly pro shirt underneath and a t-shirt. And, wear long undies and cloth track pants and you should be good-to-go in most weather. You'll be running a lot, which means you'll warm up, which also means you don't have to bundle up for snowmobiling, etc.