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 (definition) 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 (definition) (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.