Why do you think the Van Allen belts are dangerous? It might be if you spent a day in them but for one thing it is a BELT. Does that suggest anything to you?
It isn't a sphere of bad ions all around the Earth, it is mainly a wide band around the equator. That said, it is still dangerous if you spend too much time in it.
But think how long it takes to get through when they are going 7 miles per second, the speed it takes to escape Earth.
I don't know the exact extent of the belts (there are more than one) but at 25,000 miles per hour, or around 40,000 klicks/hour, in one tenth of an hour you go 2500 miles so even if you plow through the thickest most dangerous part, that is 5 or so minutes and you are through. ISS deliberately orbits below the belts because those astronauts live there for years and that WOULD kill you if you were stupid enough to live in the Van Allen belts that long. As it turns out, NASA engineers are not stupid.
As to batteries, they had Nicads back then, not as powerful as our Lithium Ion ones but good enough, don't forget, they didn't make two hour long movies, just a few minutes here and there.
I don't know how they dealt with temperatures on the moon, maybe just shielding, since it would only be hot if exposed to the sun. Also, again, they didn't spend days there, just a few hours.
Do you know when air conditioning was invented? Like 1790 or so using ammonia as the coolant fluid. 1960 technology was not as primitive as you seem to want to think. They used active cooling in the space suits of the day, remember, they were not designed to be used for weeks on end, but HOURS only.
The main thing they didn't have back then, advanced computers.
The computer onboard the lunar lander was based on verbs and nouns with just a bunch of push buttons to activate some program feature, a verb being an action and a noun being a designator, so Verb button push, (activate thruster) Noun button, for 10 seconds, and that was to start say program 31, subroutines in the computer but there were not many of those, a few dozen, maybe a hundred.
Just the radar returns coming in too fast was enough to overwhelm the computer and forced them to go to manual control the last mile down or so.
That lack of strong computer tech is what really ticks me off about the deniers, like they think somehow some film guru can come up with the incredible detail of the moon and Earth from space. NOBODY could have just drawn that iconic Earthrise photo by Apollo 10 coming around the back side of the moon and seeing Earth for the first time from that perspective.
Also the retroreflectors left on the moon had to be pointed fairly accurately at Earth so powerful lasers could send a beam to the mirrors which bounced the beam exactly back in the direction the laser beam came from. So a million watt (pulsed laser, low average power) sends a huge batch of photons to the moon and if the beam is aimed off by just a mile or so it misses the reflector pac completely and there is zero return of detectable photons.
Only if the beam is right on target do you get just a few identifiable photons that can be used to calculate the exact distance to the moon, within an inch or so and they have shown directly how much the moon recedes from Earth each year, not much BTW, maybe an inch or two. That measurement was 100% FLAT IMPOSSIBLE without the retroreflector placed there by humans. No robot tech of that day could have come even close to doing that under some kind of remote control since there WAS no such tech back then, they had the little rover car but that had no remote capability, just an electric golf cart with an open canopy and some batteries that would drive for a few miles at most.
That alone shows humans had to have been there to plant that retroreflectror aimed just right to enable laser beams to go to the moon and back with phase control intact which is what allows the calculation of the moon's distance.