Even assuming the ice is pure water, wouldn't it be radioactive given that some molecule come from the sun and the moon is exposed to radiations with no protective atmosphere?
Hydrogen from the sun is generally not deuterium, let alone tritium. Irradiation does not, as a rule, make the irradiated thing radioactive; the hydrogen nuclei aka protons are "radiation" because they're moving very fast and cause damage when they hit things, not because they're inherently different from other protons.
Being exposed to radiation does not make something radioactive. Radioactive particles settling on the thing is what usually causes things to become radioactive.
Correct, the lunar soil is itself radioactive because it's subjected to neutron radiation. What's more, the radioactivity of lunar soil increases with depth; the Apollo 17 Lunar Neutron Probe Experiment found that the radioactivity peaks at about 1 meter under the surface. I believe this is because the lunar soil moderates the incoming high energy neutrons (thermal neutrons are more likely to be absorbed.) This effect means that careful planning would be required when shielding a Moon colony. If you just haphazardly bulldoze lunar soil onto your habitat, you could easily make your problem worse.
Also we could do some approximations how much radiation does the atmosphere block? Even at factor of passing 1 to million or billion. We would still have huge amount of radioactive water in our seas and glaciers. But we don't.
