[Graham Boak]

The key point is that the codes are not permanently attached to the unit, but to the carrier, and at different times will be carried by different units.

The carrier each had an identifying letter, as you describe. Eagle was E. Formidable was F, but was seen (later) with the symbol of the Greek letter PHI (=F).

The second number identifies the role, and the seniority of the squadron carrying out that role. By 1939 that meant torpedo spotter-reconnaissance 1-5, fighters 6-7, but the earlier numbers had been used for the spotter-recce role only, which is why Swordfish units tend to carry 4 and 5.

The final letter is that of the individual aircraft in the squadron.

The first, carrier identity, code was often omitted because the RN was rarely in a position to operate more than one carrier in any given force. Hence the codes you mention last.

For a full list, you need a copy of the Air Britain book Squadrons of the Fleet Air Arm. All the codes you list are correct, 824 being E5.
800 A6, 801 U6 then A7, 803 A7 then S6 then A7 and A8, 804 uncoded with Gladiators, 813 E4, 819 L5, 821 A5, 823 G4

The S code was for shore-based operational squadrons.