How many are there in Mi-na's family?

There are five in her family: her father, her mother, her older brother Min-su, her younger brother Min-ho, and her.

How many people are there in Mi-na's family?
How many members are there in Mi-na's family?
How many people are in Mi-na's family?
How many members are in Mi-na's family?
How many are Mi-na's family?
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Is the answer to the first question correct?
What about the other questions? Are all the questions grammatical?

Absolutely!

P.S. For the last I'd add "How many are in Mi-na's family?"

Yes, the answer to the first question is correct.

Regarding the other questions, they are all grammatically correct. These questions essentially ask the same thing but are phrased differently. The word "how" is used to inquire about the number or quantity of people in Mi-na's family, while the phrases "there are," "in her family," and "Mi-na's family" provide the context for the question.