Flexibility completely depends upon the airline. One cannot speak for all airlines, only their own. At FedEx, we get 12 sick days per year, if unused, carries over to subsequent years. If you get sick or injured, you use your sick bank, if you run out, you go on long term disability (60% pay for two years). After that, you go to 50% pay, and I think it's on a max salary of 265-270K, something like that.
As far as unpaid time off, sure. You can take that 2-3 months off to climb Mt. Everest as long as you get your 3 takeoffs and landings in 90 days. So go do a 3 legger, steal or beg for all the landings, and then take the time off. If you don't get your 3 and 3, you have to go to the sim, which isn't the end of the world. The caveat is, they have to have enough staffing in your seat position to do it (which can be tough in December), or you can give away your trips. Many airplanes are well staffed. You can use your vacation to get large blocks of time off, paid. Some people like to take the summer off.
But most people here don't use the flexibility to take large amounts of time off, though some do periodically. Most people pick up extra trips, by choice. It's nice to have the option, should one ever get in good enough shape to climb Mt. Everest!