Is the woman sinful for refusing to leave the house with her parents—who get angry because she wears the isdal—and is it appropriate to inform the mosque's sheikh about this, knowing that he is a friend of her father?

As long as the issue concerns the hijab and not the niqab, do not fulfill your parents' request to go out without a hijab. You bear no sin for not fulfilling their request, for obedience is only in what is good and permissible, and there is no obedience to a created being in disobedience to the Creator.

As for the issue of informing the mosque's sheikh about this matter, we do not see it as a suitable idea, for fear of embarrassing your father, unless you believe that the Imam can influence your father and convince him. In that case, the idea can be conveyed to him without informing him of your father's stance.