What is the ruling on a male teacher giving a private lesson to a girl in her home in the presence of her family, while adhering to Sharia guidelines? And is this better than the girl attending mixed-gender lessons outside the home?

The default is for women to teach girls. If that is not possible, a man may teach a group of girls, provided that there is no fear of temptation (fitnah) and that Sharia regulations are observed.

Similarly, it is permissible for a man to teach a single girl in the presence of a mahram, while observing Sharia regulations. This is better than the girl going to mixed groups due to the potential harms that may result.

It should also be noted that a girl in middle school or high school is often an adult (baligha), so she must observe proper covering (sitr) and stay away from temptations.