The offer is good as you will be getting free accommodation which is the main expensive commodity in Qatar. First check with the HR department if you child is covered for school fees. School are very expensive as well. Your HR status should be family even as a single parent. If you don’t’ apply for this then you probably won’t get the family status benefit from the company. For example children school fees, medical insurance etc. Check out the fees of the American school for estimates.
http://www.asd.edu.qa/
http://www.acs-schools.com/doha.aspx
As for the fiancé , it would be better to get married to bring her on the family visa. Otherwise she can stay as a visitor for maximum 6 months. And renter Qatar for another 6 months.