Whether or not the mammogram is cloudy has nothing to do with the woman still having periods. If the mammogram is cloudy or not conclusive then they do an ultrasound. If cancer cells are forming but there is not a visible lump, this would mean that they just started forming. That is why it's important to get screened at regular intervals. First, if you've had previous mammograms the person reading the mammogram can probably tell because of the difference. Second, if the cells weren't visible this time, they will be the next time.
Yes, excessive radiation over a lifetime can increase your risk of cancer. However, mammograms and necessary x-rays, as well as CAT scans and MRIs do not increase your risk of cancer because the amount of radiation used is small. This does mean that one should not have these procedures without a medical reason to do so.
Because excessive radiation can increase the risk of cancer, those who administer these procedures have to protect themselves from exposure. But the ordinary person does not need to be concerned.
Some people who worked in high radiation places, such as Hanford, WN where components for the atomic bomb were made have developed cancers. So have those exposed to nuclear accidents such as Chernobyl also have developed cancers. These are massive doses of radiation and not what we receive with a mammogram.