What is the youngest age allowed to fly any sort of plane in the USA?
The easiest way to figure this out for the US is to look at 14 CFR Part 61 


  §61.83   Eligibility requirements for student pilots. To be eligible
  for a student pilot certificate, an applicant must:
  
  (a) Be at least 16 years of age for other than the operation of a
  glider or balloon.
  
  (b) Be at least 14 years of age for the operation of a glider or
  balloon.


That is the age at which you can solo. Lots of airplane pilots take lessons and solo on their 16th birthday. I don’t follow the gilder world and balloon world, but I assume lots of kids solo on their 14th birthday.

To get a license you must be a bit older:


  61.103   Eligibility requirements: General. To be eligible for a private pilot certificate, a person must:
  
  (a) Be at least 17 years of age for a rating in other than a glider or
  balloon.
  
  (b) Be at least 16 years of age for a rating in a glider or balloon.