When we first see the Fairy Godmother, she is disguised as an old beggar woman who asks Ella for some spare bread or milk. Some folklore has people leaving such things out for the local fey or household spirits.
In general, it's a stock trope of fairy tales for a downtrodden protagonist to help a stranger in need, as this usually results in the stranger revealing something that can help said hero/heroine. So the Fairy Godmother's actions are likely an allusion to this.
This idea may be a reference to another Perrault fairy tale called "Diamonds and Toads" where a kind young girl shares her drinking water with an old beggar woman and receives a precious gift, whereas her less generous sister ends up with a curse.
The "first branch that brushes your shoulder on your journey" is an allusion to the Grimm brothers' version (which was used in Into the Woods), where it was planted and turned into a tree that housed Cinderella's mother and later gave Cinderella her gown and other such things for the ball.
Why is the Godmother disguised as an old beggar woman? To make it a true Secret Test of Character for Ella. Due to the Beauty Equals Goodness trope, some people might help a beautiful person out just because they find them attractive. And likewise if she appeared to be somewhat rich, people might help her out in the hopes of getting a financial reward. Disguising herself as an old beggar ensured that someone who didn't expect to gain anything would help her - purely out of the kindness of their heart.
This version finally gives an explanation as to why Cinderella's slippers don't disappear at midnight with her gown, carriage, and coachmen: because they were created by magic, whereas everything else was made by changing pre-existing things. As the fairy godmother's Exact Words go, "Remember, the magic will only last so long. With the last echo of the last bell at the last stroke of midnight, the spell will be broken, and all will return to what it was before." The coachmen changed back into animals, the carriage returned to being a pumpkin, and the ballgown again was a torn old dress, but since the slippers in this movie weren't something else beforehand, they couldn't change back.
Ella's deep sympathy for Kit when he admits that the King is dying has an extra layer of significance when you remember that her own beloved father is dead- and that he fell ill while travelling, so Ella never had the opportunity to say goodbye. Ella pausing while running from the palace to tell the King that she hopes he knows how much his son loves him, even when she's rapidly running out time to get away before the magic wears off, might be because of this: since she never had the opportunity to remind her father how much she loved him before he died, she at least wants to make certain that the King does know how much Kit loves him.
The scene in which Kit agrees to marry Princess Chelina if Ella can't be found takes on this when you realize that the Captain no doubt told Kit about the Grand Duke's plan. It was a test for the Grand Duke, one which he failed the moment he urged the Captain to walk away from Ella's house.
This would also explain why Kit just "happened" to ride out with them.
Does create a bit of fridge logic for the Grand Duke. If your plan is so secret, why did you tell the Prince's best friend??
Could be a simple case of arrogance; certain figures of authority assume that people will obey them because of their rank and never expect that personal relationships can trump respect for certain positions.
And power-hungry schemers tend to assume that anyone else close to a figure of authority got there by being a power-hungry schemer, too. It may never even have occurred to the Grand Duke that the Captain's loyalty to Kit wasn't just a facade to keep in the young royal's good favor.
The very old song "Lavender's Blue", sung by Ella's mother early in the movie and then sung by Ella while the Grand Duke's search party is at her house, is well-chosen for its appropriate symbology. Some lyrics below do appear in the movie, but even those that don't appear are purposefully relevant:
Lavender's blue, dilly dilly, lavender's green
When I am king, dilly dilly, you shall be queen
Who told you so, dilly dilly, who told you so?
'Twas my own heart, dilly dilly, that told me so
The king and queen references are obvious, as well as the (unsung in the movie) reference to "my own heart" (that is, Kit's heart) telling him so.
Lavender's green, dilly dilly, lavender's blue
If you love me, dilly dilly, I shall love you
Let the birds sing, dilly dilly, and the lambs play
We shall be safe, dilly dilly, out of harm's way
This verse echoes Ella's and Kit's later conversation, where each gently asks the other to accept them for who they are. And to help drive the point home, Kit is wearing a green coat while Ella wears a blue dress.
Why didn't Cinderella just take a needle and thread and sew up the torn ribbons on her dress and go to the ball?
She's been through a whole lot of shit before that, all while keeping up a happy face. This was simply the last straw — even though, at face value, it's a simple problem that could easily be fixed, from Ella's point of view, it's the latest in a years-long Trauma Conga Line, and she's finally hit the point where all she can bring herself to do is cry. It was probably the fact that it was her mother's dress, combined with the fact that her stepmother refused to let her have this one bit of happiness and fun, that pushed her over the edge, but after enough time enduring all that abuse, anything could've finally pushed her to her breaking point.
It was less about the dress and more about what it represented. She had been possibly downplaying her stepfamily's abuse for a while ("they treat me as well as they are able") and still felt sorry for the two sisters. In her mind she had maybe thought that they would at least let her come with them, particularly when she had said she wasn't even looking for the prince (which was true as far as Ella knew) and therefore wouldn't interfere in their plans. Ripping the dress was the wake-up call when she realised her stepfamily truly hated her, she has no other family left in the world and would forever be without happiness.
Even if she did sew up the torn ribbons and somehow made it seem like a brand new dress instead of a patch up work, how would she go to the ball and avoid being recognized by her stepfamily?
It would take her at least an hour, maybe more, and then she'd have to walk to the ball, or ride a horse if any were left in the stables and arrive at the ball looking like she, you know, just madly rode a horse along dirt roads to get to the palace. Sweaty, dusty, muddy, very late, and in a patched-up dress is not an ideal way to show up at a royal ball trying to re-introduce yourself to your missed connection in the woods.
When the stepmother is trying to blackmail the archduke, why doesn't he just, you know, have her thrown in prison to silence her rather than agree to her terms? The more, uh, permanent solution, obviously would be too dark for this movie.
Maybe because he doesn't know whom else she could have told - and if she's thrown in prison she could easily talk to someone else and give them the information. And really, advantageous marriages for her two daughters and giving her a noblewoman's rank isn't that hard, particularly when her demands will allow his own plans to unfold accordingly.
Because the "mystery girl" is a potential monkey wrench in his plans, which are reasonably well-intentioned if not exactly in line with the prince/king's. Lady Tremaine offers a path to a great deal more certainty. Even if it was unlikely the mystery girl would show up, it wasn't impossible - but Lady Tremaine is there to help him ensure it becomes so. And as previously mentioned, she's not asking much for a man in his position. Also, he's not evil per se, just working at cross-purposes for what he believes is the good of the realm.
