All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
A wife must obey her husband if he calls her to bed and it is forbidden for her to refuse unless she has a sound religious reason like being in menses, being sick, and the like. House work and other things that are mentioned in the question are not a valid reason. It is confirmed that the Prophet said: "If a husband calls his wife (to have sex with her) then she must respond even if she were (working) at the furnace." [At-Tirmithi and Ibn Hibbaan] This means that even if she is cooking bread on the furnace, as it is impossible to leave the bread on the furnace and go elsewhere to do something else before this task of cooking bread is finished. Even so the Prophet ordered her to respond to her husband. Being shy because she lives in the house of her husband's family, in not a sound excuse to disobey her husband.
As regards delaying Ghusl (having a ritual bath to remove major impurity), this is not permissible, as this causes delaying the prayers from their prescribed fixed times. Delaying them is one of the great major sins. Therefore, she immediately has to perform Ghusl to perform the prayers after her husband finished fulfilled his desire from her.
Going out, or eating, or carrying any other duty while being in a state of major impurity is permissible as long as that does not cause delaying the prayers from their prescribed fixed times. However, it is a Sunnah for a person in a state of greater impurity to have ablution before eating as 'Aa'ishah narrated that the Prophet used to have ablution, as if for the prayer, when he wanted to eat.
Finally, it should be noted that it is not permissible for a woman to show her hair in front of her brothers in-law as they are marriageable men in relation to her and she has to wear the correct Hijab in front of them. Allaah says (which means): {…and not to show off their adornment except only that which is apparent (like palms of hands or one eye or both eyes for necessity to see the way, or outer dress like veil, gloves, head-cover, apron, etc.)}[24:31]. For more benefit, please refer to Fatwa: 84729.
Allaah knows best.