Is it permissible for a man to propose to a woman during her `Iddah(waiting period)?

It is impermissible to propose to a woman in her `Iddah, but there is no harm in making an implication to such an end in case she was observing it due to the death of her husband.

Is wearing a cap permissible for boys, or girls?

In principle, anything is permissible as 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

Generally speaking, I`m dutiful to my mother and opt for obeying her. However, she chose a certain girl to be my future wife, but I`m not pleased with her choice although that girl comes from a good family and enjoys good character. Should I obey my mother or choose another girl that lives up to my expectations?

Praise be to Allah the Lord of the Worlds.

One must obey their parents in whatever is in compliance with the teachings of Sharia and within one`s capacity. However, you aren`t to blame for not marrying that girl, but it is better that you tell your mother about that and it is preferable that you perform Istikharah prayer. And Allah The Almighty 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.