Can I pee with menstrual cup?

Yes you can.

Urine flows from bladder through urethra and exits through the urethral opening. Menstrual flow comes from uterus through cervix. Learn more about anatomy and check out pictures here: https://www.lunette.com/pages/anatomy-101

So yes: you can urinate and poop with the cup in. Just make sure the cup stays in place (especially after pooping).