'Does fluorine has more s-character?' is ungrammatical. That is because when you make questions with does, this does takes up on itself all the grammar categories of the predicate verb — the present tense, the 3rd person, singular, leaving the predicate verb stripped of all the categories, in the infinitive form.
He speaks English. — Does he speak English?
Thus, 'Does fluorine have more s-character?' is absolutely correct. If it sounds "unnatural" to you, you can express possession in a different way, for example:
Has fluorine got more s-character?
Does fluorine possess more s-character?
or even
Has fluorine more s-character?
which is the classical way to say it in the Modern English, although in the recent years such usage has declined drastically.