Does seeking independence in a separate home conflict with a husband's dutifulness to his father, and is he obligated to provide for his married brother, his sister, and his nephew?

What you did, uncovering your hijab in front of your husband's relatives (such as his brother, his nephew, his uncle, his male cousin, his sister's son, and so on) is haram (forbidden). In fact, it is a major sin that necessitates repentance, promptly wearing the hijab, frequent seeking of forgiveness, and performing righteous deeds. The prohibition becomes even more severe when the uncovering is in front of the husband's relatives, because they have access to and familiarity with the woman that others do not. The Prophet, peace and blessings be upon him, warned against the hamu (male in-law) by saying: "The hamu is death."

As for your mention of having your own private home with your husband, this is your right, and you should demand it from your husband if you are living with his brothers and nephews in the same place. The husband must grant you this, and his living independently with you in a private home is not considered an act of disobedience to his father.

Regarding a man's expenditure on his brothers and nephews, financial support (nafaqa) is obligatory for every heir-relative under specific conditions: that the relative is poor, having no wealth or income; that the person on whom the financial support is incumbent has surplus wealth beyond his and his children's needs; and that the one providing the support is an heir.