Also, do you need additional training to do fire calls? I know here in Chicago they have the CFD which controls both the fire dept and ambulance servies, and they pretty much only hire paramedics and firemen, and apparently it takes quite a while to get accepted. Is LA anything like that or totally different?
LA is very much like that, except that it's bigger and a bit more complicated. There are about ten million people in Los Angeles County, living in 88 cities as well as unincorporated areas. In all of these areas, at least that I'm aware of, fire departments handle the first-in response for 911 medical calls (and fire suppression, obviously). However, not all departments do their own transport. LA City FD, which handles the four million residents of the City of Los Angeles, has its own ambulances and transports its own patients, as do Pasadena, Burbank, Glendale, Beverly Hills, and others. On the other hand, the LA County Fire Dept., which handles all unincorporated areas and many cities (under contract), sends paramedic "squads" to medical calls for assessment and treatment only, and uses private ambulance services to transport their patients. They used to contract solely with AMR, but a couple years ago split the contract into five districts and AMR got only one. Some cities that have their own fire departments also don't transport; off the top of my head I know of Santa Monica, Torrance, and Pomona.
So, you can work for private companies that contract with FD for 911 transport, and then you can be involved in 911 calls, but you won't be handling the calls by yourself, and generally the medics will be responsible for all the assessment. Your role will be limited to BLS treatment, if that, and transport. You don't need to be a firefighter or have anything beyond EMT-B for this. On the other hand, yes, to be a firefighter, it takes years and you generally need to be a paramedic to have a shot at getting hired these days, especially if you're male.
I think I'll probably end up wanting to do 24 hour shifts since I'll be in school, seems like it might work better, though that depends I suppose.
As noted above, these shifts tend to wander through the week, although 24 on/48 off is not typical in LA. Most LA departments follow the LA City schedule, with three rotation shifts A, B, and C, each on/off three times, then off four days (see
https://www.lafirecu.org/calendars/2007/2007-2008.jpg ).
Hope this helps.