Former big-jet airline pilot here …
I’ll try to make fairly short answers to each of the OP’s questions rather than a single magnum opus.
“Avionics” can refer to anything electronic on the aircraft. That includes instruments, autopilots, radios, navigation computers, and on some of the newer aircraft, electronic flight controls.
In general, avionics can’t be rebooted in flight. There’re no controls for that, and for critical components, no way to interrupt all power to them.
For older aircraft and less critical components, like an auxiliary navigation radio receiver, there may well be an on/off switch. For something like that, a common inflight troubleshooting technique would be to turn it off for a couple of minutes & turn it back on.
But that largely went away with the advent of digital systems; their interactions and failure modes are just too complex to let the pilots be monkeying around with switching parts off and on.
The term “electronic flight controls” generally refers to a system design where the pilot’s controls are connected to a computer, and the computer is actually the only thing connected to the physical control surfaces (ruudder, ailerons, horizontal tail, etc).
Airbus aircraft from the A320 and after are built that way. If all the EFC computers quit, the aircraft is a very expensive unguided dart & will be utterly unresponsive to the pilot’s inputs. It will then either impact the Earth or break up in flight in less than a minute.
This same sort of system is common in fighter-type aircraft, pioneered by the US F-16 and used in the F-18 and all subsequent US fighters as well as late-model European & Russian equipment.
The upside of that system is the airplance can have very safe & easy-to-fly handling qualities even if the actual aerodynamics are less than ideal. An aircraft designed for maximum economy (airliner) or maximum manueverablility (fighter) will generally be a bitch to fly. A full-authority EFC will bridge that gap, and provide a good machine for the mission and also a good experieince for the crew & pax.
Conversely, Boeing aicraft use EFC as an adjunct to the pilots’ control of the machine, not as an intermediary. The pilots’ wheels & pedals are connected by braided steel cables to hydraulic valves in the wings and tail which move to direct hydraulic power to push the control surfaces against the airflow. This is the same systems design used back into the 1950s.
The innovation is that the EFC can also push & pull those same cables to assist the pilots by improving the apparent handling qualities of the airplane. It can also push against the pilots if HAL thinks they’re doing something dumb. But Boeing’s design philosophy is to resist the pilots if need be, but never to override them. If worst comes to worst, the pilots’ inputs will carry the day.
And if all the EFCs quit, the pilots can still fly the beast at what will seem to the passengers like completely normal behavior. The crew workload will be higher and there may be some areas, like extreme high or low speeds that would be best avoided, but for typical flight regimes the loss of EFC in a Boeing is an annoyance, not a catastrophe.
Naturally, on all models from all manufacturers, EFC systems are heavily armored. That software and hardware is reliable like nothing you’ll ever see on a desktop. it also has multiple redundant independent power supplies to ensure the juice never runs out.
So, bottom line, NO, a spot of turbulence is NOT the pilots rebooting part of the avionics.