My cat has peed on the sofa. Will playing with her at/near the sofa reinforce or prevent her from choosing that location again for peeing? My intuition is that if she develops a happy association with the area, she will avoid soiling the area because it's where she goes to play.
Not necessarily. While it's true cats don't like to play where they use the restroom, you may find the cat just won't play with you on the couch. Cats use smell to find where they use the bathroom. This is why you hear that if a cat pees on a rug, it will keep peeing on the rug.
I would recommend two things. First is a deodorizer to get rid of the smell of urine, so the cat doesn't associate the couch with a litterbox. Also, there are scents sold in pet stores to attract your cat to using a litterbox (normally sold with kitten items). Playing with the cat there definitely helps, just make sure to deodorize it first so the cat will actually play there.