Learn Finnish: Kysyykö hän?

This is because of vowel harmony. The word kysyä (to ask) contains the front vowel y. Suffixes attached to front-vowel words must also use front vowels, so we use the front-vowel version of the question ending, which is -kö, instead of the back-vowel version -ko.

Why does the verb kysyä change to kysyy?

To conjugate a verb for hän (he/she) in the present tense, you take the verb stem and double the last vowel. The stem of kysyä is kysy-, so doubling the last vowel gives us kysyy.

Why does the verb come first in this sentence?

In Finnish, to make a yes/no question, you attach the question ending (-ko or -kö) to the word you are asking about and move that word to the beginning of the sentence. Since we are asking if he is doing the asking, the verb kysyy gets the -kö ending and goes first.

How do we know if hän means "he" or "she" here?

You cannot tell from this sentence alone! The Finnish pronoun hän is completely gender-neutral and can mean either "he" or "she". You would need more context from the surrounding conversation to know for sure.