Snow tires, or all seasons, are all that are generally needed. I use Cooper all-terrains on my Cherokee, and while I have chains, I haven't used them once since I bought them several years ago.
Yes, if you live in a house with plumbing through outside walls, or with poor insulation, you should leave your faucets open a crack to drip. This isn't to keep the water flowing, it is to allow any ice building up to have some place to push the water. Ice forming in the pipe will take up more space than the water it is made from, so it will expand linearly along the pipe and cause water pressure to build up if you don't give that trapped water some place to flow. It isn't the ice that breaks the pipes, it is the pressurized water.
If your roof is poorly insulated or poorly ventilated, it will tend to form ice dams. What happens is the heat flowing to the shingles melts the snow, and flows down to the edge. At the edge, it contacts the open air and freezes, forming an ice dam. This dam backs the water that is still flowing down up under the shingles, and can soak into the roof interior, ruining insulation, sheet rock, etc. This snow will melt a lot around chimneys, sky lights, and above where can lighting is mounted in the ceiling of the top floor.
If this happens, you need to get a snow rake with long extendable handles and rake the snow off the roof.
Also, remove the gutters in the fall, as they will tend to fill up with ice and exacerbate the problem.
Snow shoveling is a pain in the butt. A lot of people hire a local person with a plow on their pickup to plow their driveway. Most people have a snow blower or snow thrower of some sort. Important features are a tall auger to handle deep snow and the ability to throw snow a good distance. You also want your snow blower to have powered wheels. Whether you get this as a walk behind unit, or a setup that you can mount on your riding lawn tractor doesn't matter much. Keep the battery charged.
Modern homes do not always lend themselves to wood burning of any volume. Modern energy efficient design seeks to minimize the draftiness of a home, but fuel burning devices need significant airflow. If your home has one or two fireplaces plus a wood stove, you may find that you can't run all of these at one time. One of the chimneys or the stove will develop a draft to feed the fires burning elsewhere. State of the art devices like Renai propane heaters or pelletized wood stoves try to bring cold air into the stove from the outside for burning. There are wood stoves and "zero clearance fire places" that use only outside air and do not create the concerned listed.
Salt or sand? A perrenial question for New Hampshire motorists. The New Hampshire highway department likes salt, as do many motorists, because it doesn't require cleaning the roads in the spring and doesn't cause pitting of the vehicle paint job. Problem is that if your car is insufficiently undercoated, it will corrode. Plastic body technologies are helping to alleviate some of this, but suspension parts are still metal, as is the exhaust system and other components like brakes. Salt is also bad for the environment. Salted snowmelt by the roads can cause trees to die and interfere with wetlands animals like frogs and salamanders.
You can buy salt at your local hardware store in several varieties: mined rock salt (sodium chloride) or synthetics like potassium chloride or calcium chloride. Salt is useless in areas where water will pool up, it melts through the ice to the bottom, and the top freezes back over, and is generally more slick than it was originally. However it is really good for helping to break up ice enough to chop the rest of it away.
While it isn't good for the roads, I recommend sand for driveways. It absorbs sunlight, so it does some melting, but more importantly it provides traction on top of the ice that remains. Just be sure to have a good scrubby door mat to scrub sand out of your shoe soles.