Is it permissible to pray with a heater (stove) in front of me?
There is no harm in praying with a heater in front of you, as long as you do not intend to venerate the fire or imitate those who worship it. And Allah Knows Best.
Is it permissible to offer Salah (prayer) on behalf of the sick who is on his/her death bed, and is incapable of offering it?
It`s impermissible to offer Salah on behalf of the sick, or the dead.
Is it permissible for the father to give the Zakat of his money to his poor married son?
It is permissible for the father to give part of the Zakat to his poor son, if the latter was sound and healthy, because in this case the father isn't obliged to spend on his young son, even though; he is considered one of the Muslims poor. 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.