It can't be THAT cold in NM. These cars will start well into the -40C's. Your problem is either the battery, starter, or glowplugs.
If the battery's dead, it's pretty obvious. You'll get a solenoid click, but it won't turn over and your dash lights will dim. These cars need a good battery to start as they use quite a few amps to turn over. A worn battery will be mostly water and will freeze solid at sub-zero temperatures, making it pretty much useless.
If the starter's dead you have good voltage and amperage everywhere, but the starter won't engage or it won't spin fast enough to start the engine. Check and clean the connections. The ones at the battery and the ones at the ground are very important. A little cold might create enough resistance to cause the starter not to turn over fast enough.
Bad glowplugs will give you a check engine light.
Also, when the car is cold the timing gets retarded which makes the engine noise a little nasty sometimes. Totally normal.
It's not the cold, there's something wrong with your car.