Preparation of phenylamine/aniline
Instead of carrying out distillation under reduced pressure at the end of the procedure to remove the ether, could I have just carried out simple distillation?


You could have performed any procedure to remove the solvent:


Simple distillation  - This would be slower than what you did and could overheat the undistilled aniline causing decomposition.
Vacuum distillation - This is what you did, perhaps even on a special apparatus called a rotary evaporator. This method is still a distillation, even if your product was not distilled. Something was distilled.
Evaporation on a hot plate (or steam bath)- 



  Why is steam distillation suitable in part 3? Could I have used vacuum distillation here?


Steam distillation is suitable because the aniline is already in an aqueous suspension. If do vacuum distillation, you would need to


Separate the aniline from the aqueous mixture (extract with ether)
Remove the ether (simple distillation or reduced pressure distillation)
Vacuum distil the aniline


If you look at what you did, it is the same steps in a different order:


Distil the aniline 
Separate the aniline from the aqueous mixture (extract with ether)
Remove the ether (simple distillation or reduced pressure
distillation)


Either is probably suitable. In the end, the steam distillation is easier and less hazardous than the full-blown vacuum distillation you would need.