Is a girl's adherence to the teachings of her religion and her abandonment of mixed-gender sports with young men considered backwardness in this era?

The reference for rulings is the Quran and Sunnah, not people's opinions. Islam is a practical religion in which Allah Almighty rules without any challenger to His judgment, and the reality of backwardness lies in violating the Sacred Law. The Sacred Law has clarified the grave nature of fitnah (temptation/strife) between men and women, and for this reason, it has closed every door that leads to fitnah. A man training a woman in martial arts involves impermissible matters according to Sacred Law and causes for fitnah, though Islam does not forbid women from sports with Sacred Law guidelines. A woman must adhere to the rulings of the Sacred Law upon reaching puberty, such as prayer and hijab. Death comes suddenly, and one must hasten to comply with Allah's commands.