I had to do an hour set under those sort of conditions once. It was absolute misery: besides the guitar freaking out and needing to be tuned several times as it adjusted to the cold, I couldn't feel my fingers.
The instrument needs a chance to acclimate to the temperature. Major temperature swings can cause finish checking/cracking, and/or make inlay pop out, so give it lots of time to adjust slowly.
Don't open the case to let air circulate around it until the instrument is close to the temperature of the air; Remember, air circulating will speed up the temperature change.
The same caution is needed when returning the instrument to a normal temperature. If the instrument is cold and you move to a warm room and open the case and see the finish start to turn hazy, close the case and let it sit longer.
Amps can be cold but give tubes a while on standby to gradually warm up.
If playing indoors, I don't think I'd want to take a hot amp outside into cold air to sit either; That cold air would cause the amp's circuit board and components to go through some major thermal cycling.
As far as trying to stay warm, there are big gas-powered space heaters used to warm convention tents and such. They put out enough heat but you'd need a way of enclosing the space to keep the heat from escaping. Trying to do it with an electric-powered heater would use a lot of electricity with no benefit if the heat isn't trapped.
Personally, I would refuse that sort of situation based on my own experience. There isn't a good way to protect the instruments, there isn't a good way to keep YOU feeling good, and, all in all I think you'd put on a mediocre performance for what reason? Better than playing one song in those conditions, go serve hot-chocolate and play DJ, spinning tunes off your portable computer into a PA.