How we cite our quotes: (Chapter.Paragraph)
Quote #4
They couldn't request the services of their shoemaker even in secret. Mr. Pudding was in charge of the accounts while the King was gone, and they couldn't stir up any suspicion. (9.2)
How to dance in secret if your dancing slippers start to wear out? This is the question Azalea and her sisters face. Since they're pretty poor (especially given that they're a royal family), it's not like they have piles of cash lying around to go out and buy new shoes with, or pay for the services of someone who can mend the ones they have.
Quote #5
The handkerchief was large enough that everyone could touch a piece of it—just. Azalea spoke the promise. She had them promise not to tell anyone, or show anyone, and never let anyone know about the passage or the pavilion or the Keeper. (10.112)
Swearing on silver (like the handkerchief) is a big deal in this world, as magic exists here, so when Azalea and her sisters swear to keep the pavilion a secret, they're going to be magically bound to maintain its secrecy. They probably didn't think ahead to why deceiving others about it could turn out to be a not-so-good idea.
Quote #6
"So. If you cannot be heard from your room, then, where could you be dancing? Hmm? There are no secrets and underhanded dealings in this household, young ladies. If you are harboring a secret, then I will be told at once." (11.103)
The King is apparently not a big fan of secrets, and especially not when his young daughters are the ones keeping them. This fits with what we know about him: he seems like a pretty straightforward and honest dude, so deceit must annoy him. Add in the overprotective father instinct, and he's probably worried about his girls getting into trouble and him not being able to help them because they've deceptively kept the cause of the trouble a secret.