Is the salary the questioner receives from her job lawful or unlawful, given that she knows her relative did not intercede for her in the essential job placements, and that she did not take anyone's right?

May Allah reward you for your diligence in seeking what is lawful in earning and spending on your family. We advise you to cast away obsessive doubts and hold fast to your work. The default is to believe your father and your relative, and there is no blame upon you for taking the salary if you perform your work as required.