Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.
Is it permissible to pay the Zakah (obligatory charity) for settling the debt of a deceased relative?
It is impermissible to pay the Zakah for settling the debt of the deceased since Zakah is paid to eligible recipients who are alive. As for the debts of the deceased, they are to be settled from the estate before dividing it amongst the heirs, but if the deceased didn`t leave an estate, then it is desirable for his/her heirs to settle those debts if they can afford them. And Allah Knows Best.
My mother is 70 years old and wishes to perform Hajj and Umrah, as she has never done so before. However, she also takes care of my ill father and is afraid of falling ill herself. What should she do?
If this is the obligatory Hajj – meaning she has not previously performed Hajj or Umrah – then it is permissible for her to go even without her husband's approval. However, she must travel with a Mahram (a male guardian) or a trustworthy group of women, and she should leave someone to care for her husband and attend to his needs. And Allah Knows Best.
Is wearing a cap permissible for boys, or girls?
All Perfect Praise is due to Allah, The Lord of The Worlds.
In principle, anything is permissible sol long as there is no Sharia evidence prohibiting it, thus wearing the cap by boys is permissible as long as it isn`t done in imitation of the disbelievers. Moreover, a Muslim woman should avoid all that which attracts the attention of men because whatever leads to the forbidden is forbidden. And Allah Knows Best.