It depends on whether or not her uterus and cervix were removed, if so then no she should not menstruate.
No. Endometritis is an infection of the lining of the uterus (the endomitrium). If you've had a hysterectomy you no longer have a uterus, thus no endomitrium. If you don't have it, it can't get...
Yes you can get syphilis after a hysterectomy.
No, menstruation is particular to mammals
You can menstruate anytime, anyplace! Just make sure you bring pads and tampons with you.
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