Just to stamp out incipient confusion, the FAA defines a "booster seat" as a carseat that does *not* have an upper-body harness built into it. (That includes high-back seats that have had the harness removed.) If there is no upper-body harness, the child cannot sit in it on the aircraft. If your seat has a 5-pt harness attached, it is not a booster in FAA terms. To use it on board the aircraft, it needs to have a sticker saying that it is certified for aircraft use.
If your child won't be sitting in the seat on board the aircraft, then yes, it counts as a carry-on. If by "booster" you mean a high-back seat that has no shoulder harness, be aware that it will *not* fit in the overhead bins, so it will have to be gate-checked or checked as baggage. Backless belt-positioning boosters will normally fit into the overheads or even under the seat in front of you (if the arms are adjustable, you should set them as low as possible so that they will fit under the seat.)
If it is not an int'l flight, you won't need any ID for kids those ages.