Is it considered sinful for a girl to live and work away from her father's home, with his consent for her travel and work, especially since she is working to help her father and family, and she is unmarried?

There is no impediment for a woman to reside or work in a Muslim country other than her father's country, provided there is safety from temptation, no disobedience to the father, and adherence to Sharia guidelines. If you meet these conditions, then you are not in error.