DT,
This is a great topic,
I congratulate you.

Many good things are said already, but I cannot agree with some of them.

Bear with me, I'll refer to the novel, as usual. Since the movie is anyway the reduced version of the novel, and sequels are either taking the facts and ideas of the novel and speculating about them, thus developing the novel's characters, or disregard the novel's information and screw the characters with a total flaw as the outcome, I don't see why shouldn't we consider the novel as the most original and impeccable source of everything related to Corleones.
So,
1. Mike-Sonny.
Sonny's succeeding the Don was by no means pre-ordained. Michael was Vito's original choice, he was goomed for the position.
“Though he(Sonny) was a great help in his father’s business, there were many who doubted that he will become the heir to it.”
“Michael had been his favorite before the war and obviously the chosen heir to run the family business when the proper moment came.”
When Clemenza is driving Kay to the Hotel and they chat, “Clemenza assured her…..that the “old man” thought Mike was the best of his sons, the one who would surely inherit the family business.”
So, Mike couldn't have any jealousy of his father's affection, he had no rivals being his favorite.
On the other end, Sonny also had nothing to resent, because Michael refused the position of his father’s successor and left it to Sonny by default. Besides, Mike was much younger, and Sonny obviously had some affection, mixed with irony, for this obstinate little college boy. Remember, when Mike appears after the Don’s shooting, he gets a hug and a kiss on cheek from Sonny, and, exactly as you’d expect such a no-nonsense person, despising sloppy sentimentality, as Mike, to act, he pushed Sonny away and said grinning: “After I get used to you beating me up, I have to put up with this?” to which Sonny answers that he was worried about Mike being in danger.
But the most interesting scene is the conversation that follows after Mike’s proposal of Solozzo hit. It’s very long and informative, I can’t type it all, so take a look at it, but here’s some lines of Sonny’s: “I always said you were the toughest one in the family, tougher than the Don himself. You were the only one who could stand off the old man. I remember you when you were a kid. What a temper you had then. Hell, you even used to fight me, and I was a lot older than you. Freddie had to beat the shit out of you at least once a week.”
And later: “Sonny paused and said softly: “But you’re a Corleone after all, you son of a bitch. And I was the only one who knew it.”
The first of these quotes begins with a significant phrase, in which Sonny, in fact, admits Michael’s superiority, despite all his kidding about him being a smart college boy etc. This makes me believe that if Mike would be chosen by Don as his successor when Sonny was still alive, he’d accept that.
2. Sonny-Tom. The closest of people. They argued, and Sonny insulted him unintentionally out of his temper, but they didn’t resent words. Just read how Tom feels when he’s told that Sonny’s killed.
And the most important characteristic: when Tom is relieved by Solozzo, we read: “ When Hagen had entered the room, Sonny had come rushing to embrace him. Michael realized with a faint twinge of jealousy that in many ways Sonny and Tom Hagen were closer than he himself could ever be to his own brother.”
This is interesting, because it’s the only actual mention of any rivalry and jealousy felt by Mike to any of the family members.

To be continued soon……