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They claim there were 2 small "silver zinc" batteries on the rover.

I seriously doubt that would power that vehicle longer than 8 minutes. Have you seen the battery bank in a modern electric car? And the "lunar rovers" were built in the late 60's. They didn't have modern battery technology -- or efficient integrated circuit-based electronics to gently consume that battery power. Everything drew more power back in the 60's and 70's. Every light bulb was a power-sucking incandescent, etc.  All the electronic gear in the diagram would have drawn WAY MORE POWER than the modern equivalent, AND such electronic equipment wouldn't have used a single milliamp less because they were on the moon "where gravity is 1/6 that of Earth".

They broadcast data with antennae! How many of you noticed how soon your phone battery goes dead when you use a lot of wifi or 3G? That's the biggest power drain on any phone -- even more than the screen.

You can see in the diagram how small the batteries were. Not must potential for power storage there.

I've already pointed out how the "statistical aberration" argument is complete nonsense, so we've leave that aside until someone addresses that argument and talk about your misunderstandings of how heat transfer works.

Air conditioning is necessary on earth because the AIR gets hot, and heats us by convection. The moon has almost no air, so a person (or object) on the moon's surface will be heated directly by radiation (direct transfer of energy from the sun's rays). This effect also occurs on earth obviously, but is highly mitigated by the atmosphere which absorbs much of the radiation.

So on the moon, the most effective form of "cooling" is to wear a reflective suit, so the sun's energy simply bounces off and never heats you up in the first place. Thence the shiny white suits.

So what's the problem here exactly?

The astronauts themselves also generated heat, being warm blooded. They can't "open a window" instead of running the A/C while they run around in their space suits. Having no air for convection is a double edged sword.

And their suits were not bouncing off 100% the suns rays or heating ability. They didn't look like mirror-men. They were just wearing white. Go out in the Texas mid-day sun sometime in white long pants and a white long sleeve shirt, with a white hat and a white shirt hanging from the hat to block your neck. Sure, you won't get cooked AS MUCH by the sun, but you will still heat up.
Then move to a well-shaded place like under a carport, and note that it wasn't the air that was warming you -- it was the sun. That's why I prefer working outside in the last hour before dark (when the air barely off the day's high, or around 95) to working in full sun at 10:00 AM when the air is only 79 degrees. It's the radiant heat that proverbially kills you.

See, as a Texan, I know much more about "hot" than you :)

P.S. Whatever I noted, above, about radiant heat experienced by a Texan, is even more true on the Moon -- as you pointed out, the Earth has an atmosphere to help mitigate the heating power of the Suns' rays, the Moon does not.