Hi. The part that goes in the frame is called the strike plate here. Some people call escutcheon plates the part that is mortised into the edge of the door and holds the strike mechanism in place. I would agree that the round thing that goes under the knob is a rose but I somehow got the impression that we were dealing with a rectangular plate about 2" x 6" that went under the knob. Those were common years ago and sometimes had a keyhole for a lock.
By the way, I had another thought on removing the plate (whatever it's called). If you took a matte/utility knife and cut the paint around the plate before you tried to remove it from the door, you would stand less of a chance of chipping the paint on the door. That's the theory anyhow and sometimes it does work. Start the scoring cuts from both ends of all sides so the knife doesn't slip and scratch the door.