Yeah, your number look good now. One way to check to see if your nitrate test is working is not to do a water change for about a week and a half or so. You should see the nitrates start to creep up to 10 ppm and such. If you are not seeing this then I am still betting your nitrate test is giving you incorrect results. In fact, one of the ways you can tell how often you need to do water changes is to monitor the nitrates as they rise in your tank. You don't want them above 20 ppm so more often than not, people will figure out how long it takes for the nitrate to reach about 15 ppm. They will then know how often to do water changes. This is the most scientific method and I would recommend you try this little test to see what happens with your tank.

Changing the carbon every 2 weeks is a bit extreme in my opinion. I know some people say it deactivates very quickly, but back when I was not so religious about water changes, I could tell when the carbon became deactivated from the smell. It usually is closer to 3-4 weeks. Changing it on a monthly basis should be fine.

Oh, and yes you will need to always need to use spring water from here on out since your tap water has too high of a pH for your fish. I'm not sure if you need to mix it. You need to look on the bottle and see if it says it is distilled. If it is then you will need to mix it. If it is not, then it probably has the trace elements you need. To be sure I would test it for kH, gH, and pH. As long as the pH is in the correct range and your kH is above 4.5 dH then you should be fine and you will not have to mix it before adding it. Let us know how it turns out.