Is contraction-like pain in the uterus in the month after having an iud placed common? how long does the pain usually last?

No: You really shouldn't feel the iud in your uterus. Sometimes the uterus will cramp if the iud has fallen down a bit such that it touches or even falls down into the endocervical canal. I see this all the time. You can go back and ask for ultrasound to check and make sure iud is at the fundus and if not, ask for remove and replace another one. It can be very uncomfortable, i'm sorry you have this.