Hi
In most instances during this period, a Midshipmans appointment required a degree of patronage, usually from a family member or friend. Midshipmen were as young as 14-16, but were sometimes entered even younger and rated as Captain's Servant.
The formal training of officers did not start until 1863, at what was to become BRNC Dartmouth. Since the Navy only provided board and lodging to Midshipmen, they were free to leave at any time. The problem with Midshipmen is that because they were not Commissioned the do not feature in the Navy List until they passed as Lieutenants, which required a minimum of 7 years service before they could sit the Board.
For the sons (and daughters) of Warrant Officers, the Greenwich Naval College was available. However, if a boy entered the navy via this route he usually entered as a Mates Assistant, having already received extensive training at Greenwich.
Re the records of your man, there is a Service Record for a Henry Charles Young, Naval Cadet in ADM 196 (available online), but the time frame is a bit loose, so it might be another man.
Martin