You would use mm. For the same reason you use feet instead of miles to measure the height of a building.
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Centimeter
Thickness is measured in units of length. You'd have to choose the one that gives a reasonable number corresponding to whatever you're measuring the thickness of. If it's the thickness of a hair, you'd probably select a sub-multiple of a meter. If it's the thickness of the atmosphere, you might select the kilometer. If it's the thickness of country gravy, that's a whole different story.
0 - 1 inch Micrometer might be better to measure 10 pages and find the average.
That depends on which aspect of the cube you are measuring and the size of the cube.. If you are measuring the dimensions, you might use meters. If you are measuring the surface area, you might use square meters. If you are measuring the volume, you might use cubic meters.
Anything larger than a millimeter and shorter than a couple of meters. It also depends on the kind of profession you are in. Metal workers basically use millimeters, so they would use 980 mm instead of 98 cm. Wood workers like to give measures in cm, they would say 45.5 cm instead of 455 mm (Millimeters). one mm = 0.1 cm one cm = 0.01 m