Imagine the corner points of the hexagon are numbered 1,3,5,7,9,11 and the mid-point of the edges are numbered 2,4,6,8,10 and 12 correctly in order between the points - pretty much like numbers on a clock.
Draw a line from mid-point 12 to mid-point 6 and then draw another line from corner point 3 to corner point 9.
This should divide the hexagon into 4 identical pieces.
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Select any point inside the hexagon and draw a line segment to any point on the boundary of the hexagon. Draw 7 more such segments. These will divide the hexagon into 8 parts. The parts will not be equal but that was not a requirement of the question.
Divide it into 72 pieces, group them into groups of 8. DoNe
Yes - divide the hexagon by drawing a line connecting two opposite sides, dividing it in half. (Drawing a line connecting two angles will only result in two congruent quadrilaterals).
Draw lines from every other angle that meet in the center.
In general it cannot be done. In the rare case that the hexagon is regular, select three alternate vertices and draw lines to the centre (centroid) of the hexagon. These will form three congruent rhombi.