1. Yes, just like for land plants. But just as with land plants, there can be too much of a good thing.
2. Definitely not as high as possible - or you'll kill the fish.
A little goes a long way. In an established tank, maintained correctly (reg water changes, filtration, resonable fish load, etc) with growing plants (or algae!), nitrate is usually not an issue. In newer tanks, until they're several months old, need to do more frequent water changes than in older, well maintained tanks.
3. Since the fish excrete ammonia, the "nitrifying bacteria" in a well established filtration system break it down quickly - first to Nitrite (less toxic to fish than ammonia, but still toxic in sufficient quantity), then to nitrate. Even then, if you don't have an established filtration system (or a large enough one for the fish load) where the bacteria have had sufficient time to colonize, the break down process will be slow. This is how so many people kill fish, by cleaning their filters too well (by using cleaners or disinfectants - clorox, etc.) and destroying all the "good" bacteria in the filter media. It also causes the whole balance of the tank to get off.