There are point and shoot cameras which expose their shutter count. For example Canon G series (G11 for sure).
Also, may I assume that 'shutter-actuations' always link to camera
that use mechanical shutter? (It does not apply to those camera that
use digital shutter?)
Please do not put two questions in one :-) Thank you. :-)
No, as you saw above there are point and shot cameras whithout mechanical shutter which expose their shutter count. And this is because not having a mechanical shutter doesn't mean that the camera isn't stressed. As you can imagine, the sensor and the lens are used a lot when the camera focuses and takes the picture. However, the sensor will last many many years before it will degrade / fail. Unfortunatelly I cannot say the same thing about the lenses especially if the lenses have a retractile design which can "eat" dust inside causing the lens to malfunction (believe me - it happens, just yesterday I had a point & shoot with problems because of this).
Hence the number of shots counts also for the point & shoot cameras, mainly because of lens' mechanics.