This is not an idiom. It means just what it says.
Temper: the personality and state of mind
Genius: a person much more intelligent than the average
I do believe you heard the idiom wrong. It's "apple of one's eyes." See the link below for the meaning.
Yes, it is an adverbial phrase. The phrase "after all" is an idiom meaning "nevertheless."
The word "genius" originates from the Latin word "genius" which referred to a guardian spirit or deity that protected an individual or place. Over time, its meaning evolved to encompass exceptional intellectual or creative abilities.
A synonym for temper is temperament.
Towering
Meaning you snapped and got angry.
To lose ones temper and react violently.
Lose your temper.
A genius in residence or resident genius is the smartest person in the room, the one everyone else always takes their questions to first.
"To be" is not an idiom - it's a verb.
It means she got really upset or angry, and lost her temper.
Pest is not an idiom. It's a word.
The idiom "apple shiner" means the teacher's pet.
The meaning of the idiom in the pink of health means being in good health.
It's not an idiom - to cope means to deal with, or to handle
The idiom means impress someone is egg on
"Old hand" is an idiom meaning having lots of experience.