Is it permissible for a woman to travel for work while residing in a nearby house for the purpose of work, and is her earnings lawful despite working one week and resting one week, and is it permissible for her to perform Umrah with this money?

If a woman travels for work without a mahram, it is not permissible unless there is a necessity or a need. Traveling without a mahram does not affect the earnings. If the manager is authorized to permit the woman to work one week on and one week off, then her earnings are lawful. However, if he is not authorized, it is not permissible for the woman to receive wages for the week she is absent from work, and she must return them. It is not permissible for a woman to perform Umrah except from lawful earnings. According to the majority of scholars, Umrah performed with ill-gotten money is valid, though sinful.