Hi Orion, great summary. However, I think we could add that plants also use ammonia and nitrites for food - which is one of the reasons cycling is so much milder in planted tanks. Plants basically use any nitrogen source (ammonia, nitrites, nitrates).
The 1-inch-per-gallon rule is definitely just a "guideline" and works for most smallish fish, but not for larger fish or messy fish - or for dwarf puffers (which need at least 3 gallons per DP). Goldfish need about 10-20 gallons each.
There is also the surface area calculation, which is good to use if stocking something like a Bi-Ube/Bi-Orb tank or a very tall tank.
(tank length x tank width) / 10 = inches of fish that can be stocked
Then there is the allowance made by having live plants, which suggests that stocking can exceed the "guidelines" by some percentage if the tank is lightly/moderately/heavily planted.
There are no set-in-stone rules, but just general guidelines that are especially helpful for newbies.