Michael was widely regarded to be a noncombatant by the different Mafia families, despite being the son of Vito Corleone, due mainly to the fact that he stayed out of the Family Business, kept his nose clean, and was a certifiable war hero. He might as well not be part of the family as far as they are concerned, when there are far more relevant targets such as Sonny (or Vito himself) whose deaths would actually affect how the family was run. Once Michael gets drawn in, he is shown to be very ruthless, and very cold. When asked what he would do after killing Sollozzo and McCluskey, he replied that he would sit down and finish his meal. When his godson is being baptized, he has his men do a mass-killing of the rival family heads and Moe Greene. In the second movie, he is told that he pissed off a lot of people when he did that, as it wasn't how things are done, and it is implied that this bought him a short term advantage at the cost of long-term problems. Why does he take this approach? Because he doesn't have any direct background in the Mafia. He is a soldier, who was trained to deal with his enemies far more directly than what is considered appropriate in the American Mafia. If someone is an enemy and is dangerous to you, you kill them before they can do the same to you... if you are talking about two armies on the battlefield.
The opening scene of The Godfather gives us Amerigo Bonasera asking Don Vito to kill (implied) the men who ravaged his daughter as vengeance, describing how "she will never be beautiful again." This scene is mirrored later in the film after Sonny's death, where we find Don Vito asking Bonasera not for vengeance, but simply for him to use his skills as an undertaker to make Vito's son beautiful again.
In Part I, Vito says to the other dons, "I swear, on the souls of my grandchildren, that I will not be the one to break the peace we've made here today." Technically, he kept his promise, but in Part III, his granddaughter Mary, an innocent civilian, is killed by a bullet meant for Michael.
Even more appropriate considering how heavily implied it is that Vito orchestrated the whole thing, with Michael just "pushing a button", so to speak.
Vito refusing to kill the two men who brutalized Amerigo's daughter seems, at first, to be a Pet the Dog moment for Vito. He says he's doing it because it wouldn't be justice. Instead, he's just going to have them badly beaten. It allows us to see Vito as a Friendly Neighborhood Gangster. However, if you take a step back, there's numerous practical reasons why Vito would do it this way. Murder is a very serious crime which runs the risk of endangering his family and soldiers on behalf of someone who is of very little social and economic importance. Two dead civilians would also draw police attention more than dead mobsters.
Amerigo is also an outsider who might suffer an attack of conscience and turn to the police. He'd then have knowledge linking Vito to two murders. By contrast, Vito having two men beaten up badly for harming a young woman is unlikely to draw nearly as much attention, especially if their crime is common knowledge. It also would be far easier for Amerigo's conscience to stomach in the long-term, even if he wants them dead now.
In addition, it's made clear in the novel that the boys come from powerful families. One's father is even a very powerful politician. Their families would be likely willing and able to exact revenge for death. A brutal beating may be seen as something they deserve.
The use of Cavalleria Rusticana in Part III becomes even more poignant when the Intermezzo plays over Mary's death, and ultimately Michael's during the ending. As the Intermezzo is only played in the middle of the Opera, it highlights the realization that Michael's death is not really the end, but simply the closing of one tragic if bloody chapter in the Corleone saga. Something that Kay voices out earlier in the film: "It never ends."
Don Vito uses his connections and influence as an actual godfather (ie a guardian) rather than as a crime lord. This is because he did not wish to stay a criminal, or to let his children take to crime; he is aware that criminal life will eventually destroy the people who stay in it too long. He himself had gone down the path of crime only because he had no other choice to survive. He sadly comments to Michael that he had hoped his family would leave the underworld life behind and achieve legitimacy, which reinforces this theory.
Listen to Tom Hagen's words very carefully when he's questioned by the Senate committee chairmen about Frank Pentageli's brother's presence at the hearing in Part II. The senators, clearly suspicious about Frank's sudden recanting of his testimony in the presence of his brother, asking Tom to clarify if Vincenzo Pentageli "knows nothing about these matters." Tom's only reply is "To my knowledge, nothing." Tom obviously knew what was going on, but it was Michael who arranged to have Vincenzo brought to the hearing from Italy. Tom (likely on purpose) was kept out of this, meaning that he was never explicitly told by Michael what his intent was in bringing Vincenzo to the hearing. Ever the legal mind, Tom cunningly found a way to not commit perjury by using Exact Words.
This may not be intentional, but Sollozzo's death could be interpreted as the death of the "classic" movie gangster from The Golden Age of Hollywood. In old, pre-Godfather gangster movies, the wiseguy gangster, who always looked, dressed and talked like Sollozzo, would get killed by the incorruptible police in the end. That's how the audience was supposed to learn that "crime doesn't pay" and keep the censors from clamping down on the movie. In the Godfather, the character designed to represent the classic gangster archetype, Sollozzo, is gunned down right alongside a distinctly corrupt police officer. And by whom? The quiet, decent war hero son of the most respected figure in the community, born into a world of family, honor, tradition, respect, etc. Basically telling the audience once and for all that this would not be like the older gangster movies.
The logo for the series, the hand holding the marionette puppet strings, might seem an odd choice for a saga about a mafia family, until you remember the quiet conversation between Vito and Michael shortly before Vito's death, about how he didn't want to be anyone's puppet, and hoped that one day Michael would be the one holding the strings.
The ending to Part III. Before the passing of Mario Puzo, he and Francis had plans to make the next Godfather as Vincent's time as the Don, and he would have gone into the drug trade. It would have ended with his death sometime in the mid-90s, after being hunted down and killed in a police shootout. Taken this account, Michael's death in '97 makes it all the more tragic since the Corleone name was by that point a worthless name because of the damage Vincent had done. Michael Corleone would have truly died broken and alone.
Technically, according to a "Corleone family tree" found in the Coppola Restoration compilation as an extra, Vincent gets gunned down in 2005 after soiling the Corleone name by getting into the drug business.
A flashback in Part II reveals that Sonny was the one who introduced Carlo to Connie. That means Sonny had to live with the knowledge that he introduced his little sister to the man who would later abuse and mistreat her. He probably regretted that for years. Sonny's anger issues aside, it's no wonder he flipped out so badly when he saw Connie had been beaten. It's likely he was as angry at himself as he was towards Carlo.
More on the flashback in Part II, it's revealed that of all his siblings, Fredo is the only one who supported Michael's decision to join the Marines. At first it looks like a heartwarming moment... But then, once we put it together with just how badly Fredo resents Michael, his father's favorite son, for stepping over him, the scene takes a much more darker turn. Of course Fredo wanted Michael to join the Marines; of course Fredo wanted Michael to keep out of family's business. Michael's mere presence makes him feel weak, ineffectual, and conscious of his shortcomings. His resentment to Michael has been growing all this time.