Hold on. Nothing is wrong. Relax.
Now to the issues. First, nitrate at 10ppm is fine, in fact very fine. Most suggest keeping nitrate below 20ppm, but many fish seem to be OK with it higher than that. But as a good practice, keep it below 20ppm, and it will be with regular partial water changes (weekly) and certainly with live plants.
As for the ammonia, Prime detoxifies ammonia by changing it to ammonium which is basically harmless. Test kits including the API read ammonia and ammonium together as "ammonia". So the "ammonia" is probably ammonium. Which is why the fish are not showing stress. Bacteria and plants use ammonium/ammonia, so no issue there.
Nitrite. This cannot be at 5 or all the fish would be dead. Nitrite at .25 is dangerous and at that level fish would be showing stress; heavy respiration, hanging at the surface in an attempt to get oxygen, listlessness, followed by death if not rectified. Nitrite gets in the bloodstream and prevents fish from carrying oxygen in the blood. I know that Prime detoxifies nitrite (and nitrate) by binding it somehow, so the bacteria can use it (the nitrite that is) but I do not know if tests will still show it. Unless something is way out of whack in the tank, you are not going to see nitrite above zero, unless it is in the source water. Have you tested your tap water for ammonia, nitrite and nitrate? You should, as knowing it will come in with fresh water during water changes is important, so a conditioner like Prime can be used and handle it.
While water changes will not harm the fish, all that work is unnecessary (unless there really is an issue, which frankly I doubt). Close observation of the fish by sitting in front of the tank for an hour or more is one of the best "tests" we can do. It is amazing what I find by just doing this. I never test for ammonia or nitrite, haven't for years; and rarely do I test nitrates because I know from what I'm looking at what they are. Fish behaviour is a tremendous guide to what is going on in the tank.
Inga, I know you are deeply concerned about providing the very best for your fish, and there is absolutely nothing wrong with that. But I would not fuss over testing after a water change; test the water before the water change (esp for pH and nitrates), that will give you an idea of what is going on; and keep a chart with the date, time of day (this is important for pH and nitrates) and the numbers, so you can identify changes. The water change plus the conditioner will obviously affect the water chemistry. If you want to check these things again, do so the next day. Unless you see trouble of course.
Byron.