Most of my winter riding is a 5 mile commute to the bike shop on back roads. I have fallen down twice in the last few years, each time on ice. Ice is frequently found at the side of the roads and occasionally in the middle if there is a low, flat spot. I have learned to be very careful when braking, that is when the wheels can lock up on ice and put you down quick.
Where and how far you ride will determine what sort of tire you need for your bicycle. If you ride a few miles around your neighborhood or on a bike path or packed gravel, you do not need high pressure tires or fat ones with lugs on it.
How do you know if your bike helmet fits? Put helmet on and bend over at the waist, putting your head toward the floor. The helmet should stay on your head, not fall on the floor. If it falls on the floor, tighten the appropriate band in the back of the helmet or insert foam pads.