Enigma of 討厭. Why does it mean "to hate"? As far as I know 寸 is a depiction of 


  a finger pointing at a wrist where one’s pulse was taken. The distance
  between a hand and that point is small, so this portion signified a
  little…




So IMHO 寸 has the double meaning of:


to measure
to press


Some say 寸 has the additional meaning of "something small" (I personally doubt that, yet).

So 討 should mean 言(to talk) + 寸(to press) = to ask, to demand.

Examples:
  1.討親  to marry a wife. 
  2.討求  ask, beg, demand, (to press on with ones desires)
  3.討情  ask for leniency. 
  4.討伐  to suppress rebels, to vindicate authority. 
  5.討飯  beg for food.

So here we can see that "to press" and "to ask" are somewhat synonyms under the roof of 寸.

But 討厭? I can't understand how "to press/ask" + "dog eating meat in cave" should form a meaning of "to hate".

Does anyone know the story behind that 討厭 word?
厭(厌) is a phono-semantic character. The "dog" part only indicate its pronunciation.
In Old Chinese, this character means "fulfilled". Then its meaning changed to "become bored, dislike". So 討厭 literally means "ask for + being disliked".