Is it possible for a passenger jet to start its APU using power from the RAT? If so, are there plausible benefits to doing so (like supplying hydraulic pressure to the wheel brakes)? Do modern passenger jets frequently (ever?) operate with the APU not running?
On the 737, the APU is typically started from battery power; this is normally done on the ground, but it could be started this way in flight as well. (The 737 doesn't have a RAT.) The APU is typically operated only on the ground for short-haul flights, although there are cases such as a generator that failed in flight when it's started & used in the air. It's also used in cases of a "bleeds off takeoff" to provide air conditioning and pressurization when the engines aren't supplying bleed air (to increase power for a hot, high, and/or heavy takeoff).
But yes, it's more common than not for modern passenger jets to fly with the APU off on moderately short legs.
The exception to this is that some jets are required to run the APU during long over-water (ETOPS) legs, so that if one generator fails, the APU's generator is available without the question of "will the APU start after having been cold-soaked at altitude for so long?" So in that case running it for much of the flight is common.
As far as the APU vs a RAT, the APU is almost always going to be a bigger generator than a RAT, so it can power more electrical systems on the aircraft than a RAT. These would probably include electric-motor driven hydraulic pumps, which again will probably have more capacity to provide hydraulic power to flight controls & other things than a RAT would alone. Plus, an APU can provide pneumatic power (bleed air, i.e. pressurization) than a RAT can't. The other side of that coin is, a RAT is typically able to produce power more immediately after it's called on than an APU, which takes some tens of seconds to start up.