The reason jet nozzles change diameter is due to efficiency. A jet (or rocket) nozzle is most efficient if the pressure of the exiting gas is equal to ambient air pressure. If the pressures aren't equal, then some thrust will be wasted. This is why rockets have large bell nozzles, to allow the high pressure exhaust gasses to expand and thus lose pressure until it matches outside air pressure as closely as possible.
Variable-geometry nozzles are most often found on afterburning engines. Using the afterburner results in an increase in exhaust gas pressure, so the nozzle needs to expand in order to allow the pressure to drop to match ambient.
Unfortunately, the answers to a lot of your questions are going to be different from engine to engine. I'm not familiar with the specific one you asked about, and a quick Google search didn't turn up anything useful. Nevertheless, I'll answer what I know.
If the engine is turned off, does the nozzle "reset" to a resting or neutral position? Or does it just stay how it was when the engine is switched off?
Since there is no point in moving the nozzle when the engine is not producing thrust, I strongly doubt that there will be any kind of resting or reset position for a given engine. That having been said, if the mechanism isn't perfectly balanced, then residual hydraulic pressure (or just gravity) may push the nozzle into either the fully open or fully closed position over time.
Can the nozzles be moved by hand?
Most such nozzles are hydraulically operated, so whether or not you can move them by hand depends on whether or not the hydraulics are pressurized. Some may be moved by an electric motor turning a jackscrew, and since reverse-driving a jackscrew is nearly impossible, the answer would be "no" in that case.
Does the nozzles' dilation change if the aircraft is left idle for long periods of time?
The diameter of the nozzle depends on the difference between the exit gas pressure of the engine (which may be measured directly or estimated from the engine's current power setting) and ambient air pressure. If neither of these are changing, then there is no reason to change the nozzle.