Is it permissible for a sick person to pray while sitting on a chair?
All perfect praise be to Allah, The Lord of the Worlds It is permissible for a sick person, who is unable to stand up, to offer prayer while sitting on a chair. If he/she was unable to prostrate while on the chair, the he/she should pray while sitting on the ground, but if he/she was neither able to bow (Roku),nor to prostrate, then he can do them while sitting on the chair. And Allah Knows Best.
Is it valid for the 'aqīqah to be performed using the newborn child's own wealth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for the guardian to perform the 'aqīqah using the newborn child's own wealth, because the 'aqīqah is a voluntary act of giving (tabarru'), and a guardian is prohibited from making voluntary expenditures from the child's wealth. Should he do so, he becomes financially liable for what he spent. And Allah Almighty knows best.
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.
Can an Udhiyah be made up if its time is missed?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
If the sacrificial animal (udhiyah) is a voluntary (nafl) offering and its prescribed time is missed after the three days of Tashriq have ended, then it is not to be made up as a sacrifice; rather, it becomes merely a sheep for meat.
However, if it was a vowed (nadhr) sacrifice and its time is missed, then it must still be slaughtered, and the slaughtered animal is to be treated as it would have been during its prescribed lawful time. And Allah Almighty knows best.