Is the driver sinful if he unintentionally hears a musical tone coming from a work application, and is it permissible for him to look at the hand of a non-mahram woman when she is describing the way or paying the fare, and is this considered among the types of looking that are permissible out of necessity?

Merely hearing something unintentionally incurs no sin. One should move the phone away from the ear when there is a musical tone. As for looking at the hand, it is permissible to the extent of need, such as for transactions, giving testimony, or teaching. It is advisable to lower one's gaze as much as possible.