Yes, yes. Mutations can be caused by radiation, chemical mutagens, copying errors, etc.
No, not all mutants in the MU are defaulted to readiation exposure. The exact details have varied wildly from writer to writer, as they so often do in comics.
Yes, "Children of the Atom" does arrise from the atomic fears of the cold war, so on and so forth.
From a genetic standpoint, the original mutation pretty much has to occur in their anscestors, whether that's the parent(s), or futher back. Most mutations would only result in a few altered cells over the life of the original victim. However, if the mutation occurs in the reproductive cells, then it can be passed on and will be present in all cells of any descendants who inherit the mutation.
BTW, my tone with Vic has been far more pleasant than my gut reaction to someone who is blatantly mistaken and just keeps talking.