Impact of Transformer Ratio to Performance of Transformer
Will this impact the transformer performance? (ex. saturation level,
  peak current).


All other things being equal.....


The transformer with the higher number of turns on the primary will be less likely to saturate its core. This is because the magnetization inductance is greater.
The peak current is usually defined by the load and, more turns means higher leakage inductance and resistance hence, the transformer with the least number of turns will handle load current more effectively.



  Will this secondary RMS current 3.42A is able to go up until 8.58A
  without saturation?


Saturation is due to primary current


  If this power supply is suddenly loaded with 3.7A for short duration,
  can this transformer handle it without saturation (same question as
  2)


Saturation is due to primary current