The correct sentence is:
2. Don't ask people too many personal questions.
Explanation:
"Too much" is used with uncountable nouns (e.g., "too much water").
"Too many" is used with countable nouns (e.g., "too many books").
Since "questions" is a countable noun, "too many" is correct.
Examples:
Countable nouns: "too many apples," "too many options."
Uncountable nouns: "too much coffee," "too much information."