A few comments/suggestions...
I'm using the API 5 in 1 test strip, and the API liquid one for ammonia.
You really need to get liquid test kits for everything. In my experience, those 5-1 test strips are good for detecting "0.0" of things, but the minute you have a trace of what it is you're trying to watch for, they're really bad.
I just did another test today, nitrate and nitrite readings are 0.5 and 20ppm, the color is really hard to tell, it's more like between 0 - 0.5 and 0-20. PH is 8.5.
As others have said, it sounds like the tank is still cycling. Any nitrIte readings other than zero definitely means its still cycling.
This is a reef + fish tank so I think there has to be some ammonia in it.
A cycled tank, no matter what type, should normally not test positive for any ammonia. Unless something has died, and is decomposing, you really shouldn't be seeing any ammonia.
I have been doing partial water changes for 20% every 3-4 days for the past 2 weeks.
This is a good thing since it appears you're still cycling, and have things living in it. Keeping the ammonia levels down will increase your odds that things survive, but will prolong the cycle.
I buy the water from a LFS down the street from my house, I brought sample of their water and my tank water to another LFS which i buy fish from, they said the water is very good, and my water is cycled.
Probably just being picky, but the water isn't what's cycled... it's the tank. Very little bacteria lives in the water, but instead lives in the rocks and the sand. There really isn't such a thing as "cycled water." I think all the LFS was saying was that there wasn't any ammonia or nitrites in the water. They would have no way of knowing if the tank was cycled or not because they do not know the history of the water parameters. I can give a LFS water from a cycled tank, and water freshly made that has never even been in a tank, and it can look identical.
I did not do anything to kill them, I just keep cleaning the tank glass with a sponge.
Is this a sponge that is dedicated to your tank and not used anywhere else? And it was new before you used it in the tank? Just wanting to point out that you want to be really careful about cross contamination of things in your tank. Also that many sponges these days are coated/sprayed with various chemicals to retard bacteria growth, fungal growth, etc. You don't want this stuff in your tank.