Not always because the diagonals of a rectangle bisect each other but they are not perpendicular to each other.
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any isosceles triangle
With a straight-edge and a compass:Swing arcs from each end of the segment with the compass (without changing the settings)Connect the intersections of these arcs.The resultant is a perpendicular bisector of the segment.
Can't be sure what you're asking, but it would be two circles with equal radii longer that half the length of the segment, using the endpoints as the origins of the two circles.
No. Since a line is infinite, it has no mid-point. A bisector must go through a midpoint so nothing can bisect a line (not even a segment).
With a compass and a straight edge and the lines must bisect each other at 90 degrees