Should existing drywall and floor be cut to install a partition wall
You're asking a simple question and I'll provide a simple answer but in order to meet building codes this may get a little complicated for you.
Yes- You will need to remove drywall and attach the framing of the new wall to the existing framing. There shouldn't be a need to cut through the subfloor. The baseplate or footer of the framed wall can attach directly to the subfloor.
However, beyond that it gets more complex. You will need  to get a permit and check your local codes. Constructing a separate room will need to meet codes regarding ingress and egress, electrical (light switches, outlets) and possibly other requirements such as a closet if you want it to be classified as a bedroom. You will also need to think about heating and cooling which may require adding ductwork or baseboards. In other words it's not as simple as just building a wall.