Is the husband entitled to drive his wife's private car to transport his family, and does the wife have the right to prevent him from doing so?

Your use of your wife's car and driving it in a way that you did not both agree upon, or without her consent, is considered an تجاوز (transgression) and ظلم (injustice). You must refrain from driving it unless for a fee or with her approval for your personal needs, such as running your errands or transporting your family.