Is Zakah (obligatory charity) due on the Zakah money received by a poor person, and reached a Nissab (minimum amount liable for Zakah), and a whole lunar year had lapsed over having it in his possession?

Yes, the poor who possessed a Nissab for a whole lunar year is obliged to pay the Zakah due on that money even if it was given to him as a Zakah money in the first place. And Allah Knows Best.

There is a job opportunity in the Nuns Hospital. Is it allowed for me to take this job given the difficult living conditions and the high rate of unemployment?

All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

If your work is lawful and has no effect on your beliefs, then there is no sin on you and do your best to represent the true face of Islam. And Allah The Almighty Knows Best.

Is it permissible for the mother to give her children from the Zakah (obligatory charity) of her money?

It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.

Is it permissible for a woman who is in her confinement to fast upon seeing blood signs that are neither black, nor red ?

Brownish, reddish, and yellowish discharges are all considered confinement until there is no colored discharge which marks the end of confinement. Moreover, the aforesaid woman is prohibited from fasting until she attains ritual purity. And Allah Knows Best.