Is it permissible for a veiled woman who could not find work due to her hijab to accept a job at a company that requires removing the hijab, or to take out an interest-based loan to start her own business, knowing that this is the only way to support herself and her mother under difficult circumstances?

It is not permissible to remove the hijab for the sake of work. One must be patient and search for work that does not conflict with the hijab. As for borrowing with interest, it is forbidden (riba) and is not permissible except in a case of extreme necessity or a need that is considered equivalent to necessity and is difficult to endure.