A fire Captain does not have the authority to fire employees. Such a decision would be made by the fire Chief. The Captain would document the misconduct of the firefighter and in all likelihood he would be fired or at the very least suspended for his gross misconduct involving fire equipment.
While the dispatcher is on the phone with the little girl during the home invasion, they state that the phone does not have global positioning system (GPS) capabilities. The phone is clearly a smart-phone mobile phone which universally have GPS capabilities. Even if turned off to conserve battery or avoid detection, calls to 911 automatically turn GPS back on.
Emergency dispatchers are trained to answer every call with "Where is your emergency?" in order to determine location in the event of a disconnect. The dispatcher does not get to determining location until several questions into the phone calls.
In the pilot episode, the camera is positioned behind a glass wall where conversation is occurring. As the glass door closes, the reflection of the camera and cameraman are visible when the door comes back into frame.