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.  

Is it permissible to move a grave from its current location to another place, for example, from a garden in a home to a cemetery, knowing that the grave has been there for a long time, about fifty years?

It is not permissible to move the deceased from his grave, as this may result in breaking his bones and violating his sanctity. Such actions are not allowed. And Allah Knows Best.

What is the ruling on swearing a false oath by the Holy Quran?

Swearing a false oath by the Holy Quran dips the oath-taker in Hellfire, and one who had done so should turn to Allah in repentance, seek His forgiveness, give back rights to whom they belong, and pay the oath expiation.

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.