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.
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 = 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.
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. 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 would she avoid being recognized by her stepfamily?
Maybe because he doesn't know who 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.