I donated blood today and just removed the band. Is it normal for the hand and arm to feel cold? This is my first time and I donated at 12:20 pm today

Yes. That can occur even in frequent donors, and may last for a few hours. Sometimes it is from irritation of a nerve near the vein and sometimes from the time held in extension. First is just "there" until gone, second will get better if warm arm. Thanks for donating!
Did . Did the band feel particularly tight? It may have restricted your circulation. If your symptoms have persisted, or if you have any bruising or numbness then please be seen immediately.