'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 *speak**s*** English. — *Do**es*** 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.