I got stitched up at the ER because I cut my hand, when do I have to get the stitches out?

It depends. It depends on where and how big the cut was. The er should have given you instructions on how to care for the area and when the sutures should be removed, based on your injury. Most simple lacerations have healed enough after 7-10 days to have the sutures taken out.
It depends. The purpose of the sutures is to keep the skin edges well approximated and to allow both ends to "solidify" together. Once, removed, the skin will not be held tightly anymore. It would depend on the depth of the cut, the length of the cut and how it is healing. If there is some swelling, it is possible they should be kept in longer as removal could cause the wound to open. Follow-up 7-10 days.