Do I have to staple romex to the studs behind the wall?
It's not up to code to leave the wire free. Stapling it to the center of the stud ensures that a nail for something like a picture won't hit the wire (as long as you know this code and don't try to drive a long nail right next to a stud). No one but yourself is likely to ever know this is wrong, and given the age of the home, it's sure to have a lot of other code violations under today's codes since they change over time.

So to answer your question, my recommendation is to install the staples near the top plate, the J-box, and every 3 feet or so in the middle. But I wouldn't lose much sleep if I heard that someone didn't do this in your situation. Just make sure you don't have any hidden junction boxes or connections made outside of a box since that would be a much bigger issue. And also, don't overload any circuits by adding too many outlets.