Most devices seem to be characterised over -40°C to ≥85°C. What limits them to cold temperatures? Can an IC be damaged by keeping it too cold? Does this apply to other devices, e.g. diodes, transistors?
Damage to an IC package at low temperatures while unpowered would be due to mechanical effects; differences in thermal expansion coefficients between the epoxy, lead-frame, and die.
Problems with operation would be due to increased resistance (semiconductors' temperature coefficient of resistance is negative). When the temperature and doping concentration is low enough, semiconductors will essentially become insulators and not conduct at all, causing unspecified operation.
Some ICs will operate just fine at cryogenic temperatures but they must start up warm to allow bandgap voltage references to boot.
In theory if some transistor "fails" due to carrier freezeout, the IC could damage itself elsewhere (not very likely, as most failure modes are thermal, and everything on the die is very tightly coupled.)
See the tutorial pages here for more.
As you note, most devices are characterized between usually -40 °C to +85 °C. Nothing says they will not work down to cryogenic temperatures.
You can characterize parts yourself below -40, and mechanical failures can be largely avoided if the temperature cycle is slow.
Some package options work, some don't. heheh. you have to do that experiment yourself.
You can characterize parts below 0C yourself ( easily using a domestic freezer.)
Astronomers just love dunking stuff in liquid nitrogen to get rid of thermal noise in their camera chips and A/D converters.
For extreme conditions, fit heaters to significant parts (large caps, problem IC's.)
Then your power sequencing systems turns the heaters on till the parts are in a temperature range you've characterized.
Other than the physical aspects of cold silicon, -40/85C tends to fit the most stringent conditions that most folks would need (commercial/industrial).
Practically, characterizing a device is a very time consuming process because it requires test fixtures and other equipment capable of the temperature range. It's not about buying a better freezer since many devices are characterized using the same test equipment used for production testing. The fun part is collecting and parsing the characterization data just to realize that the test fixture froze over and started collecting garbage data.