Good advice in the thread, and it sounds like you had a solid foundation when setting up the tank (RODI, etc.)
One piece of advice I have is rearrange your rocks. You want as an objective in the rearrangement to have as little rock touching sand as possible. Anywhere that rock touches sand is a liability, because nutrients/detritus can settle there. You have flat pieces of rock, which makes it easy to pile them up, but really, try turning them on their sides, on their ends, do something neat that creates some cool overhangs and pass-throughs. Think of how neat it would be if/when you have fishes swimming through looking for amphipods in the crevices.
Also, API is fine for ammonia, nitrite, nitrate. For salinity, keep your eyes out for a cheap refractometer. They can be had used for $20-30, and new about $50-120, they last forever, are easy to use, and most importantly are very accurate.
Early on in the hobby, you shouldn't really be running carbon, and honestly, I would just remove the skimmer and put it in the "backup supplies/failed experiments" basket. That particular model (unless it has changed in the 15 years they've been selling it) isn't very good, and is likely just a distraction from fixing real problems in your tank. You should be focusing on doing bi-weekly water changes for the first year or so in the hobby, until you really have developed an understanding for fishes and corals and their demands, from a biological and chemical standpoint. And skimming on a 10g tank isn't a very useful endeavor. It's hard enough trying to skim 29g. 10g or less, I wouldn't bother IMO.
If you are having algae problems, water changes and let your cycle go for a month or two, and then after a month or two you can add GFO products (TLF Phosban, ROWAPhos, etc) to absorb phosphates.
I would also move the brain to the rockwork, and closer to the top of the tank until you're more familiar with the brain coral, it's needs, and your lighting setup.