Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
What is the ruling on a vowed animal sacrifice?
A vowed animal sacrifice is to be distributed amongst the poor and needy, and neither the vow-maker, nor those supported by him are to eat from it.
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
Is it permissible for a woman to wear underwear while performing Umrah?
Yes, it is permissible for a woman to wear underwear while performing Umrah. This is because she should keep her regular clothes that cover all her body while being in a state of Ihram (ritual consecration) for Hajj, or Umrah. However, she should uncover her face and hands, but it is permissible for her to let her head-covering garment drape from her head down over her face when non-Mahram (i.e., marriageable) men pass by her. And Allah Knows Best.