selection of transformer for flyback converter
The PA3856.006NL can be used as a flyback transformer. Note 2 in the data sheet states this: -


  
  For flyback topology applications, it is necessary to ensure that the transformer will not saturate in the application. The peak flux
  density (Bpk) should remain below 2700Gauss. To calculate the peak
  flux density use the following formula: Bpk (Gauss) = K1_Factor *
  Ipk(A)
  


Given that the peak primary current is calculated as 10 amps on the lowest input volts, Bpk (Gauss) = 41.7 x 10 = 417 gauss and this is well short of 2700 gauss.


  Now, is my approach correct ?


Your approach does seem correct but you forgot to mention operating frequency however, i reasoned that the on time is 4.72 us if that helps.


  where is the turns ratio specified ?


The turns ratio is 0.47:1 as shown by the little numbers on the picture of the transformer in your question.


  Also, what is the actual advantage of have a dual output in the
  secondary ? Does it have to do with the current on the output and its
  effects on the transformer saturation current.


Dual output means you can wire them in series or in parallel. Saturation of the core is caused by primary current reaching too high a value.