Is it permissible for a woman who is observing `Iddah (waiting period) due to the death of her husband to travel for performing Umrah (Minor Hajj)?

A woman observing `Iddah of a revocable (Rajee`) divorce isn`t allowed to travel for Umrah except with the consent of her husband.

Is it incumbent on a wife whose husband hasn`t gone to bed with her(for sexual intercourse) for many years to observe `Iddah(waiting period) upon his death?

Upon the death of the husband, a wife is obliged to observe the `Iddah even if he hadn`t slept with her because she inherits him.

Should one obey his mother even if she was wrong?

Obeying one`s mother is mandatory in matters that are beneficial to her and as long as she drive her children to commit sin.

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.