There is no place on Earth or even under it that is "safe" from cosmic radiation. Neutrinos produced in the cores of stars including the Sun pass right through the Earth all the time as if it was not there. Go down to be bottom of a mine and they are still there, and still passing through you. However they do no damage.
On the surface we are constantly being struck by solar and cosmic radiation which the atmosphere and even the buildings we live in cannot stop entirely. This is mainly electromagnetic radiation of very high frequency, higher than X-rays. But all electromagnetic radiation is absorbed by any substance it passes through, even air, the amount of absorption depends on the exact frequency and is inversely proportional to the thickness of the medium. So if you double the thickness of the medium, you halve the amount of radiation. This applies to radio waves, infra read, light, ultra-violet, X rays, gamma rays, cosmic rays, the lot. The practical effect is that people in airliners and those who live at high altitudes get more of these than people living near sea level.
At very high altitudes, just into space are the Van Allen belts discovered years before the Moon missions. These are regions where charged particles from the solar wind produced and expelled by the Sun circulate under the influence of the Earth's magnetic field. These are not like the high frequency electromagnetic radiation I mentioned before and can be deflected or partially stopped by thin sheets of metal, plastic, even paper, depending on exactly what form of radiation it is. Metals are best. When these particles are slowed down or stopped, by the metal skin, they radiate in the electromagnetic spectrum at various wavelengths and in all directions. So the energy that these things contain is spread over a sphere and little of it would get through to the insulation inside a space capsule, then it has to get through the instruments in the capsule, then through the clothing the astronauts wear. Of course, not everything is stopped by these things, but it does reduce them a lot.
Now the other thing is the Van Allen belts are not uniformly dangerous. Some parts of them are more dangerous than others, but these are moderately thin layers compared to the entire belts. So if you are in a space capsule travelling at high speed you pass through them in several minutes. Can you pass your hand through the flame of a camp fire quickly and not get hurt?
It would take a gigantic magnesium flare to be seen from the Earth against bright moonlight. This would be like trying to see the flame of a match by day from a quarter mile away. What was done is that large amounts of radio signals were sent. Telemetry from the capsules, astronaut voice communication and television pictures. Some of this was picked up by an amateur radio operator in North Carolina with equipment he built himself. If an amateur could do this in 1969, then what could professional radio engineers in any country in the world do?
You can be very certain that the Union of Soviet Socialist Republics watched every step very carefully by radio and received the TV pictures directly, and tracked the signals across the sky to be sure they were coming from the Moon.
I don't know where you got the idea of the pressure in the suits being higher than in a football from.
This would be the reverse of a good idea. At sea level on Earth the average atmospheric pressure is about 14 pounds per square inch, and the air contains 20.93% oxygen, 78.1 % nitrogen, 0.94 % argon, a little CO2 and other gases. If you make up a special gas mixture, which is easy. you can reduce the pressure inside a space suit or space capsule from 14psi down to 7psi, as long as you double the amount of oxygen. You could reduce it still further to maybe 3 or 4 psi if you use a mixture of about 82% oxygen and 18% nitrogen. It is the partial pressure of oxygen that humans need, not the total. This would solve a lot of engineering problems.