A few comments:
Since this is the backpack hunting thread, I assume that weight is an issue.
For water purification, I don't bother with a filter any more. I use chlorine dioxide tablets, like these:
Katadyn Micropur
Unlike iodine or other water purification chemicals, chlorine dioxide breaks down completely and doesn't leave a bad taste in the water.
Not only are these tablets much lighter than a water purifier, but water purifiers are vulnerable to cold. In freezing temps, the filtration matrix freezes and breaks, rendering your water purifier useless.
As to what stove: Again, as this is the backpack hunting thread, my assumption is that you are not planning on doing any gourmet cooking and will be eating freeze dried meals and other things that only require hot water to prepare.
A butane/propane canister stove is ideal for these purposes. Others here have recommended the Jetboil. The Jetboil is a good stove, but it performs very poorly in windy conditions. The MSR Windburner or the MSR Reactor have superior performance in wind. If you're hunting in the mountains, you can expect wind.
As for how to keep your canister stove performing well in sub freezing conditions:
Buy this bowl. It weighs less than 2 ounces. Take it with you on the trip.
Titanium Bowl
When you are ready to start cooking, warm the fuel canister up under your coat before you begin cooking. Once it's warm, attach it to the stove, fire the stove up, and warm up about a 1/4 cup of water. When the water is warm, pour it in the titanium bowl and set the fuel canister (still attached to the stove) in the warm water. Start cooking.
This keeps the butane/propane fuel warm and gassified while you are heating up water for your meal. This is a much simpler, lighter, more effective, and safer method than constructing a home made copper heat exchanger or trying to keep the canister warm with a lighter.
I have used this method (with a MSR Reactor) well below zero F, with consistently good results.