Can my wife put her plastic hairbrush in the dishwasher on the top rack without problem?!?

No...especially if there is a drying cycle that is used. That might melt the frame, bristles, or both depending on the plastic in the brush.

No. The hair can clog the dishwasher's sump. And the very high temperatures of the water may cause the brush to warp if it is made out of plastic

No, that doesn't seem like a good idea.

Does it say "dishwasher safe"? If not, then I wouldn't risk the big mess of melted hairbrush in the dishwasher. 

A good way to clean a hair brush is to comb it to remove the hair and other debris then soak it in a solution of water and baking soda. If the bristles are quite oily, it may be necessary to shampoo and rinse the hairbrush before the baking soda soak. 

Soaking isn't good for wooden brushes, for natural fiber bristles. Give natural brushes a quick shampoo with tepid water, rinse thoroughly under running water (not hot!), shake off excess water, and let air dry. 

If she washes it with other things they might get hair on them.

Possibly, but why? Seems sort of gross to me, or does she think that dishwasher actually sanitize everything?

yeah, i mean u can put plastic dishes in the dishwasher, why not a plastic hairbrush?