- Does that mean the flywheel (which is connected to the engine) rotates at 800 revolution per minute while in neutral?
Yes. The flywheel is directly connected to the engine so it will rotate at exactly the same speed as the engine.
- If so, how come it is safe to fully release the clutch while in neutral?
During normal running operation (going down the road), the transmission allows power to be transmitted from the engine to the drive wheels. When in neutral, there is no power transmission. The input shaft (which is connected to the clutch plate [aka: friction disk], which is connected via the pressure plate to the flywheel) is allow to spin freely. The input shaft is not directly connected through gearing in the transmission to the output shaft.
- if I fully release the clutch (while in neutral), then the clutch disk (which is connected to the wheels) and the flywheel (which is connected to the engine) will engage together, right?
When the transmission is in neutral, the clutch disk isn't connect to the wheels. But yes, when you release the clutch pedal, the friction disk, pressure plate, and flywheel (in essence) become a single unit and spin together at the same rate. This is regardless of what gear you're in.
- Why isn't the car moving even though flywheel and clutch disk are engaged (since I fully release the clutch), and there is a rotation of 800 RPM?
Because when the transmission is in neutral, there's no power being transmitted. Without power transmission, there is no vehicular movement.
What this all boils down to is this ... you have an incomplete understanding of how a transmission works. It doesn't matter if it's a manual or automatic transmission ... if the transmission is in neutral (or even park for an auto), the vehicle isn't going to move via power output of the engine (it can roll if gravity has it's way, no doubt). There's no way for the power to get to the wheels.