Is a woman entitled to doubt the Islam of her husband, who shows no interest in religion, does not learn about it, and avoids mentioning Islam in front of his family, despite five years of their marriage and the birth of their daughter?

You cannot judge your husband to be an infidel or a hypocrite merely based on what you have mentioned. If he utters the Two Testimonies (Shahadatayn) and adheres to the outward rituals of Islam, then he is a Muslim. It is not permissible to doubt his Islam merely because he abandons prayer, except if it is upon request, or due to his not learning Arabic and religious matters, especially since he fasted Ramadan. His stance with his family does not indicate doubt in his Islam or his hypocrisy. Continue to supplicate for his guidance and offer him advice kindly to encourage him to pray, learn Arabic, and the Quran, and seek help from virtuous people who have an influence on him. If a divorce or khul' takes place, custody is yours unless you remarry; then it transfers to whoever is more entitled according to the order of the jurists.