I'd like to clear up a couple things from your list.
1) Fish produce a lot ammonia from pooping and breathing, that's bad.
Just so you know, there are other sources of ammonia. Any organic matter that is dead (plant leaves, uneaten fish food, fish excrement, dead fish, etc) will decompose which involves being broken down by all sort of different bacteria. Ammonia is produced by all of this decomposition, as is CO2. Ammonia is also produced by fish respiration, as already noted.
4) While they are growing, they will make these 3 ppm thingys a day, that's good, and after the day is over it gets magically turned into nitrate, right?
The 3 ppm is simply the level of the ammonia in the water, measured in parts per million. You are adding this yourself. So 3 ppm means there are three parts of ammonia for each million parts of water. That may not seem like much, but it is deadly to fish and many other aquatic creatures too. You add sufficient ammonia to the tank of water to arrive at this level.
The bacteria that appear and take up this ammonia are
Nitrosomonas species. Once they appear, they can reproduce roughly every 12+ hours; they do this by dividing into two, what we term binary division. This assimilation of ammonia produces nitrite. Once this begins to appear, another species of bacteria, called
Nitrospira, will appear to take up the nitrite; these bacteria can reproduce (also by binary division) roughly every 32 hours. Now, both of these bacteria species will continue to multiply so long as their food is available, being ammonia or nitrite. They will not multiply beyond what is needed; if the food source dwindles, they do not die but go into a sort of suspended animation or hibernation so they are still there but inactive.
6) You remove the nitrate by water changes (which in my mind basically means ammonia) , and so the cycle continues.
The byproduct of the Nitrospira bacteria using nitrite is nitrate. This is less harmful directly than ammonia or nitrite, but still poisonous to life forms depending upon the level. This is removed by several processes; some different types of bacteria will use nitrate, nitrogen gas which then dissipates into the air is produced, some plants may use some of the nitrate--but the largest amount is removed by the aquarist with partial water changes. The level of nitrate can vary; in tanks with live plants it is normally minimal, sometimes "zero" using our aquarium test kits. It must also be kept low by not overfeeding, not overcrowding fish species/number, cleaning the filter regularly, and cleaning the substrate. Live plants also help.
7) I know I'm ready for fish when I get a sudden drop in nitrIte and a sudden raise in nitrAte, that's good, right?
It can take anywhere from a few days to several weeks for the above processes to establish. When you have ammonia and nitrite reading zero for consecutive days, and nitrate reading something, you are "cycled."