Right before Maria seeks advice from the Mother Abess about her love for the Captain, we see the nuns greet a new candidate for their abbey. Some viewers need repeat viewings to realize that the dress the candidate wore is the same one Maria later wore (earlier, Maria said that each woman who enters the abbey has to give away her "worldly clothes") when she visited the Von Trapps again.
The family looks confused when Uncle Max announces to the audience that the Von Trapps will be performing an encore, suggesting they weren't expecting to. But he was there when they performed "So Long, Farewell" at the party, and they end up performing again as result of their encore. Thus, that encore announcement may have been a Batman Gambit by Uncle Max to covertly tell them "Try to escape after leaving the stage" by planting the idea in their head through the song. Quite impressive for a guy who otherwise seemed to be The Quisling.
The Brawn Hilda's insistence on taking lots of bows, though very funny and seemingly annoying, actually gave the family more time to escape before their names were called as winners. Max plays to the audience for comedy and isn't in a rush to force her off the stage.
When the nuns sing the reprise of "Maria" during her wedding to Captain Von Trapp, the lyrics take on a whole new meaning: He is able to do what the nuns weren't, to encourage her to stay with him ("how do you make her stay and listen to all you say/how do you keep a wave upon the sand?") because she wanted to rather than out of a sense of obligation.
Part of the reason that the Captain's first rendition of "Edelweiss" is such a heartwarming moment is that it's essentially the Captain—the famously patriotic military man—expressing his love for his homeland through song, after spending the entire movie insisting that soldiers have no room for music in their lives. The Captain's defining personality traits (discipline, hardiness, patriotism, etc.) never really change, but he learns a new way to express them to others, and we learn to love him for it.
Why is Leisl and Leisl alone singing the echo to Edelweiss when the Captain first sings it? Because she is the only one old enough to remember their mother singing it with their father before she died.
Minor one, but Gretel's role in the puppet show is merely pushing the prince onstage from the bottom - while the others operate the marionettes from the top. Obviously at only five years old, she probably wouldn't be able to work the puppets, but Maria probably put her there because she'd probably want to be able to watch the show as well. This must also be why Marta controls the backdrop rather than the marionettes as well.
Maria rather expertly manipulates things to stop the children playing pranks on her, without simply snitching to their father. She guesses that they're mainly doing it to get a reaction out of her, so she instead plays a great hand in making them feel guilty for doing so - and it also serves to let them know that she's not going to tell on them to the Captain. Maria exploits the saying "you catch more flies with honey than vinegar" to win the children over.
After learning about the Captain's relationship with the Baroness, Maria gleefully declares, "So THAT'S why God sent me here! To prepare the children for their new mother!" Well, indeed, He has.
Depending on how the song is played, Rolf's eventual turn to Nazism is foreshadowed hugely by his verse in "Sixteen Going On Seventeen". The Nazis were very much focused on promoting traditional gender roles, particularly in women, and the song is essentially Rolf's attempt to get Liesl to conform to what would be generally expected of a woman under the Nazi regime.
So what happened to Max afterwards? While losing a submarine captain wasn't a massive defeat, the Nazis were infamously petty. At the very least, he probably lost his cushy job, and at worst... well, we've seen the kinds of prison camps the Nazis ran...
If you know anything about Nazi Germany and what happens to Austria after the Anschluss, you sort of have to feel sorry for the Baroness, who is demonized for trying not to get into trouble with Germany, and Rolfe, who will likely die a soldier in the war (especially if you subscribe to the idea that he was naive enough to be brainwashed into the Nazi Party, not realizing his actions would lead to the deaths of Leisl and her family). Honestly, every character villainous or otherwise (but presumed villainous for complying with the Nazis rather than escaping Von Trapp-style) is likely to face hardship or outright death in the coming war.
When the butler greets Rolf at the door near the beginning, they immediately begin talking about something clandestine. The butler looks over his shoulder and shuts the door. He and Rolf were already with the Third Reich at that point and probably were talking about the Captain being made to join as a Third Reich soldier, which of course they wouldn't want the Captain to overhear.
Rolf telling Leisl about some things about to happen in Europe:
Rolf: "Don't tell your father. He's so Austrian."
Leisl: "Well, we're all Austrian."
Rolf: "Well, some people think we should be German, and are angry at those who don't think so."
Rolf delivers another telegram and does the Nazi salute to the Georg, who becomes angry and commands Rolf to leave.
Baroness: "Oh, darling, he's just a boy."
Georg: "Yes, and I'm just an Austrian."
At the party the general Rolf mentioned earlier is there and he's angry at the Austrian flags that have been hung up. Later Georg is more rightfully angry at and destroys the Nazi flag that he finds displayed over his own doorway after his and Maria's honeymoon.