Believe it or not, us non phd's do know about black-body radiation and the S-B law.
Now, perhaps, after having been spoon-fed those concepts by people here. Earlier you were trying to compare temperatures of objects in space to temperatures in the meteorological sense, as of some congruence could be expected. That is a tell-tale rookie mistake. I've used and taught these concepts literally for decades. I'm quite familiar with the misconceptions newcomers bring to the table.
Many things within the cabin did not have to reach extreme temperatures to not function.
But you've provided no justification for the premise that anything in LM would reach any specific temperature. You've just given the layman's wrong-headed impression that vacuum would "rush in," and that this vacuum would be icy cold.
According to NASA docs, for instances, many of the electronic components needed 30 F to function.
This just commits the same error as your prior argument. Yes, it is generally possible to discover the lowest temperature at which various Apollo equipment was expected to function. You focus on that premise and leave alone entirely the premise that any part of the LM got to temperatures that made those limits an issue. You know your argument is weak on that point, so you hammer the other one distractively.
Further, it has been stated several times that electronic components generate heat simply by the fact that they are passing current. You're still arguing as if you believe there is some ambient that has an extremely low temperature, and that these components are soaking in it. As I mentioned at least twice, the temperature-sensitive equipment can be mounted in the AEB, where it receives the full force of the sun.
Which I knew many would argue prior to posting my questions.
No, you didn't.
But what about A13? They felt things getting colder not hotter.
Because the CM is an aerodynamic vehicle, its thermal design is less forgiving. It cannot eliminate all the conduction paths to the skin. But more importantly, the thermal design of all the Apollo spacecraft presumed that there would be a source of heat. I already explained this earlier, in my lengthier presentation on thermal design. You obviously didn't read it. There was no flight scenario contemplated in which all the spacecraft electronics would be shut off, depriving the cabin of its source of heat. Hence the thermal design presumed that heat source as part of computing its equilibrium temperature.
As a matter of historical fact, the cabin temperature was reasonable until the astronauts covered the windows to block the sun while they slept. Then it dropped to near freezing. Why? Because the solar influx was enough to maintain the temperature using the residual heat from before, when the electronics were operating. With the windows blocked, the equilibrium of the interior surfaces was disrupted, and that affects the rate of heat flow. The difference between influx to maintain a temperature and influx to raise to a temperature is one of the many counterintuitive things about heat transfer, and why it takes an expert to understand it properly.
So either posters here are correct, and the astronauts were lying. Or the A13 Astronauts are correct and the posters here are wrong? Which is it?
Or the ubiquitous third option: once again you don't know what you're talking about, and are manufacturing dilemmas out of your copious ignorance.