Is it permissible to work in the field of medical analysis, which requires mixing with men, in order to provide a source of income for the family, despite the availability of temporary funds for living expenses and the father's desire for that, or should one continue to refuse the work?

There is no objection to working in the field of medical analysis if it does not lead to forbidden mixing (ikhtilat) or forbidden seclusion (khalwa). If the work necessitates forbidden mixing, then it is not permissible to engage in it except out of necessity. It is obligatory to obey one's father if he commands something that has clear benefit and from which no harm is feared.