LCD screens don't like extreme temperatures and can suffer permanent damage.
Yes, but good luck finding a non-LCD handheld multimeter.
Fluke 101's storage temperature range is -30 °C to +60 °C
To beat that you'd have to go to something like a Fluke 28 II:
TemperatureOperating -15 °C to +55 °C, to -40 °C for 20 minutes when taken from 20 °C
Storage -55 °C to +85 °C (without battery) -55°C to +60°C (with battery)
But a Fluke 28 II is a
lot more expensive.
(And a lot bigger - I don't know how big your glove box is, or how full...)