However, this has nothing to do with the need to constantly "sweep" the image on the display.
Yes it has. It is the term used by hardware people, it is the term everybody uses. Get used to not being correct when you wander outside your area of knowledge.
So, sweeping and multiplexing are two completely independent/orthogonal principles here.
No. You have to use the words as they are used by the rest of the world.
the question is: do the segments get damaged by 100% duty cycle?
It depends on how they were designed. Some designs overdrive the segments in order to compensate for loss of brightness, on these designs stopping the multiplexing will burn out the display. Others are more conservatively designed and will not over drive the display. You find out what you have by consulting the data sheet for forward current and looking at the current limiting components in the design.
Or do they need 25% duty cycle to have time to "rest" and "cool down"?
Displays do not get too hot when they are over driven. It is a common beginners mistake to believe that the average rating of an LED can be equated to the continuous rating in the data sheets. Other factors like electron depletion in the conducting band of the semiconductor also come into play. If an LED has a peak rating above that of the continuous rating it will say so in the data sheet along with the duty cycle this peak current applies to.