What is the ruling on answering a phone call from a woman who fears she will harm herself if her call is not answered, and how can one get rid of her harassment, and is it permissible to use a gift she sent, with the obligation to donate its value on her behalf?

The basic principle is that speaking with a non-mahram woman is permissible if there is a need, provided it adheres to the dictates of Islamic law. If there is no need, it is not permissible. It is obligatory to limit the conversation to the extent of the need, while ensuring there is no temptation (fitna). If temptation is feared, or if her voice incites desire, it is not permissible to reply to her, and it is advised not to reply at all. As for a gift, it is desirable to accept it, unless it is a compensation for an illicit relationship, in which case it is not permissible. If it is not for such a reason, you may benefit from it, and you are not obliged to give its value in charity.