Hi,
So we're talking Mr Freeze from bats?
OK lets look at the basics of such a weapon(?) Cold can kill in a number of ways, but the normal one that gets people is hypothermia, and that requires no such cold temperatures. Just enough for the core body temperature to fall so that it no longer works. People die from being out in the woods overnight with thin clothing at temperatures well above freezing.
Your guy won't be killing through hypothermia as far as I can see. His polar blasts would actually be freezing flesh. And frozen flesh basically dies. If you've ever seen freezer burn on a steak you'll have some idea of what I'm talking about. Essentially ice crystals form in the flesh and rupture the cells, so that what is frozen becomes dead flesh. On warming a piece of freezer burned flesh you'll see that it's basically white and pale. The cells have been ruptured so that none of the liquids remain in them. Now freezer burn is similar in its effects on flesh to regular burns, in that too much damage over too much body kills. The reason, ruptured cells are lysed by certain chemicals in them. Those chemicals if released in too large an amount, will lyse healthy cells as well, leaving you with a dead man. This is a sort of toxic shock which kills many burns victims.
There's also a condition called hypothermic shock. This is what kills swimmers in cold water so very quickly. In essence the body instantly reacts / over reacts to the cold temperatures and draws blood away from the extremities leaving hands and feet white while the warm blood is kept in the body core to keep vital organs working. This strategy works well if you can get out of the cold. But unfortunately if you can't, your limbs essentially stop working. So this is your numb fingers and feet. Muscles need blood supplies to work, and if it's cut off, they don't. This is why those suffering from hypothermia often lose the ability to walk etc and can no longer make it to shelter.
Also your guy's weapon would overcome many natural defences against cold. In essence your body loses heat according to a couple of simple physics principals. In essence heat loss is proportional to the area exposed to the cold, the temperature differential, and inversely proportional to any insulation. Nature works with this, designing cold dwelling creatures to have shorter thicker limbs (More volume to surface area on the organism - see the body morphology of eskimos versus negros) and by adding insulation - blubber and fur. Polar bear fur which is remarkably insulative works by trapping a layer of air against the skin. If your weapon comes with wind that can ruffle fur, it will rob the poor old bear of his protection against the cold. In addition to which the wind means that by standing still a person can no longer be surrounded by warmer air as he would be in calm air. In essence a person standing still in calm air, radiates a little heat into the air, but the air surrounding him stays warmer. Thus he sits in a sort of insulative bubble. Blow that air away and his skin is always exposed to the freshest coldest air. This is the wind chill factor in action and why you feel the cold more in windy places.
Hope that helps.
Cheers, Greg.