The autopilot will disengage if the controls are manually manipulated while the autopilot is on. The main issue with this particular airplane was that there was no audible autopilot enunciation that indicated that the autopilot turned off which is what the pilots were used to hearing on Russian made jets. Rather there was only an indication light that would go on if autopilot disengaged and could go unnoticed. The autopilot disengagement was not noticed right away and the aircraft entered into a spiral spin. The pilots tried to recover from the spin but they pulled up too much (over corrected) and the plane entered a stall.
Autopilots can disengaged automatically for a number of reasons. For example Air France 447 autopilot disengaged due to conflicting information from the airspeed indicators. Incorrect actions by the pilots after the autopilot disengagement caused that accident. In modern day aircraft the autopilot disengagement is heard as an audible signal so the pilot knows to take over. If Aeroflot 593 had this feature the accident could have been avoided (besides not having your kids fly the airplane).
To answer your questions about he various levels of autopilots there is really only one autopilot that works in conjunction with other automated systems to fly the plane below is a description of the various systems:
Autopilot: Automatically flies the plane according to instruments. This system is responsible for manipulating the control column.
Auto-Thrust: Automatically adjust thrust to maintain a constant airspeed.
Yaw-Dampening: Automatically adjust rudders for coordinated turns. Yaw-Dampening can work when autopilot is on or when the airplane is manually flown.
Just because the autopilot disengages does not necessarily mean the other systems disengage.