Do you mean the white paper the template is drawn on? That's just the kind you put in any printer. For patterns literally anything that's flat like paper will work. Cardboard, poster board..hell you could even waste money and cut a pattern out of wood plastic or metal, though that would be dumb, but just as examples you don't NEED to use paper and thickness doesn't matter since all you're doing is tracing the cut out on the foam.
You mean using a dremmel instead of a belt sander? If so then yea, you CAN but it'd take way longer and be extremely hard to near impossible to make the blade edge even and flat. Because of how small dremmel sanding drums are it'll be next to impossible.