Given your budget, a studded tire(s) is out. The only ones worth considering are Nokians, and here in the USA they would take most of your budget. Salt can be washed off daily to minimize damage. Even with daily cleaning of the chain and gears, 6 months of sand and salt will probably require a new chain and probably a new cassette. Think of it as commuting on the beach. If that is still within your budget, go ahead. The other complication is brakes. When it is right at freezing or just below, your rims will ice up decreasing braking enormously. You may need to ride with your brakes applied continuously to prevent this. That is very tiring physically and mentally. Disc brakes avoid this problem, but if your bike isn't already equipped with them, it is out of your stated budget. HOWEVER, WHEN RIDING ON SNOW AND ICE, DO NOT USE YOUR FRONT BRAKE, THE REAR WILL IS LESS LIKELY TO SKID! By the way, even with fenders, when slush is falling from the sky and it is around freezing your gears will ice up turning your bike into a single speed. You need to change gears frequently to prevent this or you may find yourself facing a hill on your big ring and a small cog.

Clothing is less of an issue. Tops and bottom, wool is your first choice, polyester fleece second, never cotton. For the short distance you are describing, you won't get as overheated as on a longer commute, but you still need a jacket that is going to vent the sweat you will generate. I recommend that you change when you get to your destination if at all possible. If you use clipless pedals, take them off for the winter. They ice up and so do the cleats. I like riding with ordinary waterproof winter boots with a side zip and just high enough to completely cover your ankles, about 6 to 8 inches. It is useful to have a pair of rainpants/windpants. Balaclava, ski gloves or mitts, wool liner gloves, wool socks and possibly ski googles finish your wardrobe. Thrift store purchases can bring it into your budget. Become a weather fan, listen to multiple reports to plan your trip. Be sure to allow 30 minutes to dress, and another 15 minutes to turn on all your bike lights and do a safety check on your bike.

From what you've described of your trip I agree that the cheapest, safe solution is to walk on the slushy days and when the roads have not been cleared fully. Avoid riding on days when cars can not safely pass you. When it is just cold, go ahead and ride. Consider putting some money in a jar every day you can't ride, and use it next fall to buy studded tires. This is the only time I would suggest this, but for a winter beater bike, consider going to X-mart. Buy a cheap mountain bike with disc brakes to put those tires on. Then find someone to help disassemble and correctly reassemble the entire bike, making sure that all of the rotational systems are fully greased and putting on headset seals etc. It will be a heavy tank, but you will be able to stop safely. Alternatively, watch the police auctions, and pick up a bike with disc brakes that way.