How can I smooth out a poor drywall repair?
You will need to smooth it out by applying increasingly wide applications of drywall mud; depending on how rough it is it might take 2-3 coats.  You would then sand it smooth, prime and repaint the entire surface.

Unless the drywall is water damaged or has significant physical damage (doesn't look like the case), you will not have to cut out anything.

On the ceiling you might be able to get away with just repainting the entire ceiling (prime first!) as people are unlikely to get close enough to it to really notice a few imperfections. I suspect it is more noticeable in your case because of the color difference due to the paint fading at different rates.