I wouldn't use carpet. I have an almost white linoleum (or maybe its vinyl) flooring in my sewing area. It is easy to see everything on the floor, fabric scraps, thread bits, fluff, dropped pins, etc. And it has no small crevices for things to hide in. More important it's easy to clean, just a quick sweep with a broom or a damp mop. My sewing room might be messy but the floor looks fabulous.
I don't find my floor cold although I do wear slippers during the winter.