The process is:
ammonia is made by the fish as a waste product. Bacteria in the filter and on other surfaces use this ammonia as food, and their waste product is nitrite. A second species of bacteria lives in the filter and on other surfaces which uses nitrite as food, and their waste product is nitrate. So all the ammonia made by the fish as their equivalent of urine ends up as nitrate - once there enough ammonia-eating and nitrite-eating bacteria in the tank.
Since your tap water reading is different from the water company's values, put it down to our test kits not being the most accurate. It is difficult to make a really accurate nitrate tester for use in the home.
Instead, use your readings for both tap and tank nitrates. Unless a tank is heavily planted (plants use nitrate as fertiliser), the lowest a tank nitrate can be is the amount of nitrate in the tap water used to fill the tank.
When a tank is cycled it is usual to do weekly water changes to remove nitrate and other substances secreted by the fish that we can't measure. During the time between water changes the nitrate level of the tank gradually creeps up as the ammonia continually being excreted by the fish gets turned into nitrate. If there aren't enough bacteria of one or both species to make nitrate it stays constant; and if the fishkeeper is having to do daily water changes to keep ammonia and/or nitrite at safe levels, nitrate won't go up much.
The best way to use nitrate results is to follow a trend in the tank. Once your need for water changes slows down so you can go longer between them you should find the tank nitrate increasing, though the rise might be too small to measure accurately. It's once the nitrite slows down that you should start to see nitrate rising.
As for unsafe nitrate levels, it depends on who you read.
There are some species of fish that are intolerant of nitrate but reading up on fish before you buy should tell if if the fish you want are one of those species.
Some sources say that nitrate is only harmful in the hundreds ('whole-nitrate' scale) while others are now saying anything over 20 is not good.
But as a rule of thumb, once a tank is cycled, the fishkeeper should aim to keep the nitrate level from exceeding tap level plus 20, 'whole nitrate' scale. If it gets higher than that he has too many fish or he is feeding them too much (food decomposes to make ammonia which is converted to nitrate) or the water changes aren't often enough or they aren't big enough. Or a combination of those.
It's only really necessary to test the pH of water that has been left to stand. The others change hardly at all, if any at all. pH is different because water companies tend to add things to the water supply. In hard water areas they might add carbon dioxide to the water to acidify to try and prevent limescale deposits in the water mains. On standing, the carbon dioxide gasses of and the pH changes. In soft water areas they may add chemicals to increase the pH to prevent corrosion of any metal parts in the system.
Since the water in the tank has been standing a while, it is useful to know if your tapwater does change on standing, and if it does, by how much.