Is this what should be done when someone faints? Would paramedics do this? Is it standard protocol

A) if a person is in true "cardiac arrest" there is no shockable rhythm, and the defibrilator would not be able to be use; CPR would continue until the machine detects a shockable rhythm OR a stable heart rate.
B) "two fingers under the nose" isn't the best way to check for breathing; in CPR you're taught to look at the chest wall to see if it is rising and falling.

I'm a bit confused. The girl went from "passing out" to dying? And she had a fast heartbeat, but then the EMS crew arrived and they began CPR and put her on the defibrillator? If she did lose a pulse and wasn't breathing, they would absolutely perform CPR and take her to the hospital, although they would try to stabilize her before they took her to the hospital. Of course, you can only spend so long in the field before you need to just keep doing compressions all the way to the hospital and hope they can revive her there.