What mechanisms or metabolisms do animals use to survive outdoors when the temperature is 10° below zero?

Some animals survive because of layers of fat, and/or thick fur, such as my avatar, the Bactrian camel.
There is at least one type of frog which self freezes. They secrete a fluid at the first sign of frost. This starts the process. They freeze in a hard little lump. When weather warms, they thaw right out and immediately look for a mate. (Should have thought of that before the cold.)
I’m not sure how sparrows manage, but I could swear the other day I heard one say, “Jeez it’s cold!” A duck nearby became offended and said, “At least you don’t have to keep yourself watertight so your arse doesn’t freeze over!”

Some frogs and toads bury themselves, as far as I know.

The weather has been brutal the past 3 days and most of the wildlife has hunkered down someplace. The chicken carcass and squirrel I put out for the critters stayed for 2 full nights before someone took it. Usually a meal like that is gone in a few hours.
I know that birds fluff their feathers to get extra insulation. but I don’t understand how the deer survive. The snow is covering the ground. The temps are well below freezing with high winds. They need many thousands of calories to survive. What the heck are they eating? Where are they going? Where are they bedding down? I suppose I could go out with a thermal imager and look but it is just too darn nasty. Later today when the snow stops I will check the stealth cam and see which animal took the squirrel.

Sometimes animals do die from the cold in extreme temperatures. They might live in places with cold climates, but the unusual 10 below when temps almost never go below 10 above in that region can kill some of the animals.

Some animals huddle to help stay warm. They change places within the huddle so all can survive. For instance a group of cows will stand close together and the ones in the center will move to the outside to give all the cows a chance at the protective middle of the group. I would assume many mammals do this same thing, but I don’t know that much about it.

I collected the SD card from my camera and checked. At -20C, -4F, only the fox and deer were out – and one crazy human. The fox dug the frozen squirrel out of the snow and took it away. The others: skunk, opossum, racoon (note all are “fat”) were nowhere to be found. Is it possible they are living off their stored fat and were not as desperate as the fox?

My resident opossum has been scurrying back and forth just outside my basement slider. he looks very well-fed. Since I throw my dirty cat litter outside in the snow, embedded in its ground dried corn, he may well be eating that. Certainly the deer, crows, and juncos are.

I suppose they get in out of it.

I wish @gailcalled was here, so I could ask her if the possum ever snuck into her house, like no less that 5 did, at my house, over the course of 2 years.

RIP Gail.

Oh, and to answer your question, many of them go underground to survive, like the frogs. The insects probably die, after depositing their eggs safely underground, below the freeze line, where they grow and come bursting out in the spring. Unless they’re 17 year cicadas. Then they wait a while.