How does Cather's life or views relate to "Paul's Case"?
Your introduction already gives a number of helpful comparisons that can be drawn. I think it is important to remember that Cather herself lived and worked in Pittsburgh for roughly nine years, and so no doubt she would have been familiar with the dreary middle class from which Paul emerges and finds so unbearable. Cather herself was born into a rural family of pioneers, and so presumably had to struggle in life to make something of herself.
However, far more important than this link to my mind is the importance of music to both Paul and to Cather. Whilst she was studying at the University of Nebraska, Cather developed a passion for music that dominates quite a bit of her fiction. Paul, too, is enraptured by music and in particular the way that he is able to use it to transcend his circumstances. Consider the following description:
When the symphony began Paul sank into one of the rear seats with a long sigh of relief, and lost himself, as he had done before the Rico. It was not that symphonies, as such, meant anything in particular to Paul, but the first sigh of the instruments seemed to free some hilarious and potent spirit within him; something that struggled there like the genie in the bottle found by the Arab fisherman. He felt a sudden zest of life; the lights danced before his eyes and the concert hall blazed into unimaginable splendour.
There is something then about the music that allows Paul to forget his background and circumstances and evokes a spirit of Romance and freedom within him. This was certainly the case for Cather too, as part of her attraction to music was the way that it allowed her to transcend her difficulties.