"Whim" also means "an odd or fanciful or capricious idea"--this definition could describe the nature of Ha's question, but the phrase "on a whim" connects to her desire to beat her brothers; that caused the blurting of the idea that Father might really be gone (which is not as whimsical as the guess that he had shaved his head and joined a Buddhist monastery).
From the sad look on their faces I know despite their brave guesses they have begun to accept what I said on a whim.
Here, the "d" at the end of the verb "suppose" turns it into an adjective, but it is an adjective that is always used in the phrase "supposed to." Often, this leads into a point that is the opposite of what the character is supposed to do or feel.
First day back after Christmas break, I know I’m supposed to wear everything new.
tenaciously unwilling or marked by tenacious unwillingness to yield
The adjectives "stubborn" and "fluent" are used as opposites here. A Latin root makes the contrast more vivid: "fluere" means "to flow"--so the speech of a fluent person flows like water, while a stubborn person who is unwilling to yield is like a wall.
Compare with "supposed" in this list. Similarly, the verb can be used by itself (example: Mother expected smiles from us), but the adjective is always used with "to" (example: We are expected to smile). The phrase "expected to" often leads into a point where the opposite happens or the character struggles with the expectations.
As with every Tet we are expected to smile until it hurts all three first days of the year, wear all new clothes especially underneath, not sweep, not splash water, not talk back, not pout.
A pivot is an "axis consisting of a shaft supporting something that turns"--so this pivotal year is also the turning point of the author's life. Because the readers do not see the fictional Ha as an adult, the same point cannot be made about her (yet).
I hope you enjoy reading about Ha as much as I have enjoyed remembering the pivotal year in my life.