I may be able to provide some help for the flying stuff.
Yes, the FAA has regulations limiting the flight time and duty period for commercial passenger operations, flight instruction and fractional ownership operations (basically private jet clubs like NetJets).
Services provided by Air Ops like search and rescue, aerial firefighting count as part of general aviation, hence not subjected to FAA rest requirements.
Crew fatigue is a danger to flight safety whether you're a commercial pilot or not, so it's up to individual agencies to set their own duty time limit. According to this interview with a real LAFD pilot, they work a traditional 24/48 firefighter shift schedule, and there's no hard limit set up for maximum hours per flight, but they try keeping it under 6 hours for the initial phase. If the mission is expected to last much longer, they'll start calling more people in (probably why Tommy had to leave the bachelor party) and enter crew rotation. After their first 6 hours, they'll have a rest period, from 4-8 hours, before they can go into the air for another 4 hours, rinse and repeat, until the fire is out. For those large scale campaigns, crew members can be on duty for consecutive days.
For typical operations though, while your department might advise against working back-to-back shifts for 48 hours straight, it's absolutely not against any regulation. You're likely spending most of your shift on the ground on a regular day anyway.
It's not the US, and it's a special case, but air crews of the Air Greenland search and rescue unit work a whopping 2-week shift schedule. Yes, one shift lasts for 2 weeks, 24/7. It's up to individual member to decide if they're fit for flight or if they need rest.
They once worked 12 hours a day for nearly a week to evacuate the residents of a settlement afflicted by a tsunami in 2017.
Source: Vertical Mag
More information on how Tommy's shift schedule could work. The interview mentioned is 4 years old so I'm not sure if they have changed to follow the Kelley schedule now like the rest of LAFD but it should give you a better idea about how things would work at Harbour station











