Is it normal to have a sudden feeling of sadness out of the blue, randomly, a couple times a month?

Possibly. Many of us have a bad day here or there either for no reason or for reasons we've tried to ignore. With women, changes during the menstrual cycle can increase the likelihood of such abrupt but brief mood changes. If the sadness passes fairly quickly and isn't impairing your ability to function, probably nothing to worry about. If it increases or is disabling, might want to have an assessment.
Yes. As long as it lasts less than 24 hours and is not associated with any suicidal thoughts or significant impairment.