It depends on the size of the bag. A carry-on meets the dimensions for the overhead compartment, and a personal item must fit under the seat in front of you.
You need to show government-issued photo ID to fly within the US. If you don't have a passport, you can use a driver's license, state ID card, military ID, etc. If you do not have proper ID, then you need to show up early and bring whatever identification you might have. There is no guarantee that you can fly in that case.