Is it Islamically permissible to marry a girl despite her family's refusal due to sectarian differences, and through a contract not officially documented, with the knowledge of the sheikh, in order to preserve the girl's feelings towards her family?

If you wish to marry a woman but her family does not consent, then know that a marriage is not valid without a guardian (wali), based on the Prophet’s (peace and blessings be upon him) saying: "There is no marriage except with a guardian," and "Any woman who marries without the permission of her guardian, her marriage is invalid." So, if her family agrees, then proceed; otherwise, leave her, and trust that Allah will compensate you with something better. This is with the caveat that neither the woman nor her guardian should hold beliefs that are idolatrous, or bid'ah (innovations) that lead to فسق (grave sin) or apostasy.