In "The Red-Headed League," how does Sherlock Holmes contrast with Peter Jones, the police agent from Scotland Yard?
In this story, as in many of the Sherlock Holmes stories, Holmes seems so superior to the police that they barely seem to be engaged in the same profession. That said, Jones actively seeks Holmes' help. On his part, while Holmes dismisses the man's intelligence, he does praise Jones for being brave and persistent.
The larger comparison is between Holmes' at times arrogant superiority, the attitude of a detective genius, and that of the average working police man, whose mind is much more limited.
Sherlock Holmes and Peter Jones respect each other for particular reasons, but they also find flaws with the way the other man does his investigative work. We learn of Jones' opinion first, when he vouches for Holmes to Mr. Merryweather.
“'You may place considerable confidence in Mr. Holmes, sir,' said the police agent loftily. 'He has his own little methods, which are, if he won’t mind my saying so, just a little too theoretical and fantastic, but he has the makings of a detective in him. It is not too much to say that once or twice, as in that business of the Sholto murder and the Agra treasure, he has been more nearly correct than the official force.'”
Jones acknowledges that Holmes gets good results, but he won't go so far as to say his methods are in keeping with the "correct" way of solving crimes.
In Sherlock Holmes' opinion, Jones is the one who performs badly on the job. However, he acknowledges Jones' good character, bravery, and perseverance.
"'He is not a bad fellow, though an absolute imbecile in his profession. He has one positive virtue. He is as brave as a bulldog and as tenacious as a lobster if he gets his claws upon anyone. Here we are, and they are waiting for us.'"
This respect for character rather than competency is a common sentiment Holmes expresses throughout the stories, always setting them up as the bumbling idiots forever falling short when compared to the brilliance and success of Holmes.
As for your second question about Holmes' outlook on life, early in the story he says to Watson, "'You will remember that I remarked the other day...that for strange effects and extraordinary combinations we must go to life itself, which is always far more daring than any effort of the imagination.'" One might take from this statement that Holmes understands much strangeness can be found outside of crimes, and in fact can be found in every day life, because life events can sometimes be crazier than what one might dream up in one's head. He continues this seeming respect for the less exciting events of life, saying, "'You have heard me remark that the strangest and most unique things are very often connected not with the larger but with the smaller crimes, and occasionally, indeed, where there is room for doubt whether any positive crime has been committed.'" Apparently, Holmes does not even need a definite crime to solve in order to be intrigued by an occurrence.
At the end of the story, Holmes sings a very different tune. After Watson congratulates him for once again putting together the pieces of the mystery so brilliantly, this conversation takes place:
“'It saved me from ennui,” he answered, yawning. 'Alas! I already feel it closing in upon me. My life is spent in one long effort to escape from the commonplaces of existence. These little problems help me to do so.'"
Here, Holmes labels every day life as mundane, citing crime-solving as the only means of escaping such a boring prison. He needs crimes, because without them he could not put his brilliant mind to use and would be stuck in the drudgery of regular living. This view, of course, differs greatly with the one he seemed to take at the beginning of the story.
Having said this, I would like to note that the opinion of every day life Holmes expresses at the end is more in keeping with references to life in other Sherlock Holmes stories, so the apparent shift in perception is likely not influenced by anything that happens to him during this particular mystery.
One more difference between the beginning and the end of this story is what Holmes has to say about reputation. At the start, after he explains his methods to Mr. Wilson, Wilson says, "'I thought at first that you had done something clever, but I see that there was nothing in it, after all.'" In response, Holmes says, "'I begin to think, Watson...that I make a mistake in explaining. ‘Omne ignotum pro magnifico,’ you know, and my poor little reputation, such as it is, will suffer shipwreck if I am so candid.'" The latin phrase he uses means "everything unknown [is taken] as grand." Holmes will appear more brilliant if he keeps his clients in the dark about his methods. At the very end of the story, however, Holmes has another foreign phrase to deliver to Watson; L’homme c’est rien–l’oeuvre c’est tout. This translates to "the man is nothing- the work is everything, but with this logic Holmes should not care one bit about his reputation, because it is his work that should speak for him.