Would it be suitable for visiting Taiwan from 1/10/2014 to 10/10/2014? Would it be rainy season or Typhoon season?

It will not rain a lot in October and the average temperature is around 24.3C for Taipei and around 26.5C for the southern part.
October is still in typhoon season for that island, but typhoon seldom occur then, so you will not meet one if you are lucky enough.
Overall, it is a great time for visiting that island.