Are all these ways of saying 1.59 in Chinese currency correct?
元、角 are formal and 块、毛 are their colloquial counterparts respectively. Mix and match is much less common but not unheard of. 

Omitting the last unit word is also an informal usage. 

一块五十九分 is extremely uncommon if not wrong at all, but understandable. 一块五十九 is not understandable to Chinese ears without clarification.

Therefore,


  Formal:
  一元五角九分
  
  Colloquial:
  一块五毛九分
  一块五毛九
  
  Colloquial mixed with formal fragments, less common:
  一元五角九
  一块五角九分
  一块五角九
  一元五毛九分
  一元五毛九
  
  Unnatural but can be understood:
  一块五十九分
  一元五十九分
  
  Ungrammatical and cannot be understood without clarification:
  一块五十九
  一元五十九