Is it permissible for me to give my elderly neighbor and her daughter a ride home in my car if they need help and there is no man with them, and is it permissible for a girl to ride alone in a taxi or bus to go to work, university, or to run errands?

It is religiously forbidden for a man to be alone with a woman who is not a mahram to him, based on the Prophet's (PBUH) saying: "No man is alone with a woman unless there is a mahram with her," and his saying: "No man is alone with a woman but the third of them is Satan." The state of khalwa (private seclusion) is negated by the presence of another woman with the man. It is not permissible for a woman to ride in a taxi or a bus alone with the driver, as this constitutes forbidden khalwa, and Satan whispers and beautifies sin. Keeping away from such situations is safer and more preventative.