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 for a Muslim physician to specialize in gynecology obstetrics, and what is the ruling on working as such?

Some fields of knowledge are an individual duty while others are a collective one, and specializing in gynecology obstetrics is a collective duty. However, if there were female physicians to treat women, then there is no need for a male gynecology obstetrics specialist to look at women`s Awrahs (private parts) except in necessary situations since the jurisprudential maxim says:” Necessity must only be assessed and answered proportionately.” And Allah Knows Best.

Which hour coincides with the last third of the night?

This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.

Sometimes I receive a message containing a picture of someone desecrating the Noble Quran. The sender asks recipients to send that message to their contacts as an act of defending the Quran. What should I do when receiving such messages?

Praise be to Allah, the Lord of the Worlds.

You don`t have to circulate that message, because this isn`t the proper way to defend the Quran. And Allah The Almighty Knows Best.