Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?

A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.

I took a car from the Orphans' Fund Development Institution; is there usury (Riba) involved in this transaction?

If the Sharia's conditions were met, then there is no harm in this. And Allah The Exalted Knows Best.

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.

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.