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On a three dimensional basis yes it can. Fir instance, if an object is moving directly towards or away from you the angular displacement can be zero though the distance displacement changes.
Absolute value is the distance between a number and zero, so 10 would have an absolute value of ten since it's 10 away from zero, as well as -10
No. It shows zero speed. Velocity is distance/unit time. The slope of the line shows change in distance / change in time. Since distance never changes as time changes, the change in distance is zero. Alternatively, the slope of a horizontal line is zero. If zero speed is considered "constant", then yes, it does show constant speed but the speed is zero.
The absolute value is the distance between a number and zero on a number line.