Is it permissible for a female doctor to examine her neighbor using the clinic's equipment for free, and is the patient sinful for not paying the financial compensation, and what should she do?

If the clinic administration allows the female doctor to provide free examinations for some patients, there is no harm in that. Otherwise, it is not permissible to use the clinic's tools without charge, because the Prophet, peace and blessings be upon him, said: "The wealth of a Muslim person is not lawful except with his willing consent."

If the administration allows the female doctor to reduce the fee for her relatives, then you should deduct the discount percentage and pay the remainder to the clinic. You must convince the female doctor to take the fee from you and pay it to the clinic. If she does not, then you must deliver the amount to the clinic, and you are not required to inform them of the reason for their entitlement to the money, as the objective is for the money to reach them.