First off, no I do not think you are nuts for trying this now. Itll be more difficult to get motivated in the morning, but that is all. Especially if you can make the short milage this winter and work into more miles this spring.
Tires: get some good commuting tires. Theyll be more or less the same width as your tires now but without knobbys on them. Theyll make your ride smoother and easier. Price- at least 20 dollars each, if not more depending on what you get. If youre planning on using them a while then some that are more expensive are worth it in my opinion because theyll last longer and provide more flat protection. It seems like a lot but theyll last a while and compared to filling your car with gas...
Try out a lot of shops, see who you like. There is a thread on here (dig around- its a little older) titled something like "City Bikes" in which people talk about which shops they like and dislike. Right off the bat though RiverCity in Portland is big, friendly, and will have a variety of everything you need so you wont have to search in a bunch of different shops. Stop a biker at the store near where you live and ask them where they go- theyll probably know local places.
Brakes- Get them looked over. Most likely its an easy fix for relatively cheap, but when youre not the most confident on a bike having brakes that function correctly will put you more at ease.
Weight- Get a rack put on the back of your bike and a pannier to put your stuff in (if you have the money). Itll be more comfy. otherwise its up you you whether you want to carry all that weight on your back. Try it out and see how you feel. If your shoulders and back get tired youll know what you need to do.
GET A LOCK as well. Nothing will discourage you more than getting your bike stolen.
Good luck and welcome!