To help with a few of your questions.
The youngest age that a boy would be sent to the navy would be 11 or 12, though 13/14 was more common. They would be sent to a vessel whose CAPT the family perhaps knew or a relative had served aboard. The RN officer class and indeed the upper class of the maritime population was very interconnected and it would not be hard to secure a letter of introduction and recommendation for a young boy in a fleet at war and in need of bodies.
The most obvious would be for them to enter the merchant service as many did, or to survive on what money they had saved or their family had while trying to re-enter the service. The final option could be for them to enter the service of another nation, such as Cochrane and John Paul Jones both did. Things like opening a sail shop or rope makers required specialized skills which an officer was unlikely to posses sufficiently.
Admiral of the Fleet is the highest grade of officer, an O-10 in modern terms. Thus while you are right that FSL is the highest job there is in the RN it does not refer to a person's rank.
There were at the time a group of "Naval Lords" on the Admiralty board along with several civilian lords. It was during this time that the Naval Lords began dividing up the various responsibilities such as finance, supply, personnel,... These were later renamed to the various numbered Sea Lords, at one point during WW1 there were 5 SL's with various responsibilities.
Essentially it was a coming together of merit, success, political acceptability, recommendation from inside and outside the fleet, connections, and popularity which could all play a part based on who was the PM.
Hope these help! Let me know if you would like any in depth sources. And while it may be wiki, the articles on the RN are relatively good and can help if your looking to make the jump from reading fiction like Hornblower to understanding the true nature of the RN.
Edit: Reddit seems to be extremely aggressively renumbering lists...
FOUR: Kind of. They became an admiral, but only those promoted to "Rear Admiral of the Blue Squadron" would have further employment. The others were essentially compulsorily retired. I think there is an allusion to this in one of the earlier books, with Captain Hornblower musing that if he reached flag rank the admiralty could yellow him if they wanted to, he would be satisfied with admiral's half-pay.
EIGHT: He is made a Baron I think, Baron of Smallbridge, which would make him a peer and entitled to be addressed as Lord.
I also second the recommendation above to try Patrick O'Brien's Aubrey-Maturin series if you enjoyed C S Forrester.
`2. The thing about salaries of various members of the British Navy at the time was that a significant amount of it was dependent on ransoms and ship prizes-namely, capturing ships would give you a certain amount of the value of the ship as a prize. A captain would get something like 20-30% of the prize money of any ship captured by his ship, while an admiral would get something like 10-15% of the prize money of any ship captured by his fleet. For certain postings, this could result in huge amounts of money-a Spanish ship, the Hermione, was captured by a pair of British ships, and was found to be valued at over 500,000 pounds back then. This was (and is!) an absurd amount of money. Since two ships were involved, the prize would be split, and each captain would have gotten something like 60,000 pounds, which was what he would earn salarywise in something like 70-80 years.
You can see the pay of members of the British navy here: http://www.nelsonsnavy.co.uk/broadside5.html