You asked about commercial aircraft in general, so I will give an answer from that point of view.
Is there any obvious warning when auto-pilot is disengaged?
Yes, both visually as flashing lights, and aurally. Furthermore, the lights and tone does not go away until a second confirmation is received from the pilot. For example, pushing the button on the yoke / stick once will disengage the autopilot but trigger the warning. A second click is needed to silence it.
Is there any obvious warning when mode changes?
It is not a warning, as mode changes are explicitly made by the pilot. I'd still consider it "obvious" because the active mode is shown right on top of the Primary Flight Display (the instrument you should spend most of the time on).
Picture: A/P disengage button on yoke, A/P disengage warning light and A/P mode indicator circled in red.
Image source https://www.team-bhp.com/forum/attachments/commercial-vehicles/1506426d1463061690-boeing-777-pilots-review-boeing777cockpit.jpg
So what happened on First Air Flight 6560? In this particular instance though, there are a few factors (in my opinion) that contributed to the crew not noticing the autopilot state:
- The 737-200 is a rather old design. Compared to a modern cockpit where things are seen "at a glance", it requires more concentration to fly.
- The 737 series have a rather unpopular feature called Control Wheel Steering (CWS). It allows the pilot to partially disengage one axis (pitch / roll) of the autopilot by moving the yoke in that axis only. On newer models, this feature is removed since very few pilots actually use it. On the Boeing 777 for example, such yoke movement would result in A/P disengagement and the trigger of the warnings I described.
Note that this incident is quite similar to the crash of Eastern Air Lines Flight 401, where the captain accidentally bumped the yoke, causing the autopilot to switch from altitude hold mode to CWS mode in pitch. The pilots failed to notice the change in time to avoid the crash.