Is it common for periods to stop completely after having a hysterectomy?

YES!! Yes the uterus id gone so no lining to shed and no bleeding. Now if the ovaries are left in which they usually are, then you can still ovulate and gets symptoms of cramping etc.
Yes. During a period, the uterine lining sloughs off and comes out of your vagina. If there is no uterus, you can't have a period.