Is it permissible for a woman whose menses exceeded (15) days to pray, fast and do what was prohibited on her to do during the latter period?
If the menses exceeded 15 days then it is deemed menstruation, and in this case she is obliged to pray and fast and to do what was impermissible for her to do during the latter period. In addition, she is obliged to make an ablution for each obligatory prayer after it's due time and making sure that impuity is removed. Finally, your menstruation period is over (Questioner) simlar to your menstruation in the previos month.
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.
Is it permissible to repeat the Istikhara (seeking guidance) prayer, and is it a condition that one should sleep after offering it?
All perfect praise be to Allah,The Lord of The Worlds Istikhara is offered once, but it is permissible to repeat it, and sleeping after offering it is not a condition for its validity. And Allah Knows Best.
Is it allowed to shave my beard to join the army taking into consideration that I can`t find a job?
All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
The army should be joined in preparation for Jihad (Fighting in the cause of Allah), so if Jihad contradicts with growing a beard then it is allowed to shave it. And Allah The Almighty Knows Best.