I'm sorry this didn't get a reply yet!
I would turn your skimmer back on, I don't think it's what's lowing your nitrate and phosphate, and it's good for gas exchange.
The Chemi-Pure Blue does not contain Granular Ferric Oxide (GFO), which binds and reduces phosphate. Chemi-Pure Elite is the one with GFO.
I think you should just let this ride a little longer. At two months the tank is still very new and balancing out.
What happens with phosphate specifically is that it can be bound by rock and sand, so basically the your rocks are sponging up all the phosphate. This will continue until eventually an equilibrium is reached. If going forward you are still seeing that zero phosphate, you can dose a solution of sodium phosphate to bring that up:
https://www.reef2reef.com/threads/diy-phosphate-dosing-recipe.1109919/
If you're seeing zero for nitrate going forward, I think feeding more is a good first move and if that doesn't help you can dose nitrate directly or better still you can dose ammonia, which corals benefit from:
https://www.reef2reef.com/threads/diy-ammonia-dosing-for-low-nitrate-systems.987087/
https://www.reef2reef.com/threads/diy-ammonia-dosing-chart.1123913/
I don't know what you recored for ammonia, but if you're suing something like the API brand ammonia test kit, it's sometimes a little squirrely and can read like 0.25 ppm to 0.50 ppm when the concentration is likely much lower. Even if real, ammonia up to about 0.50 ppm isn't much to worry about. If it starts hitting 1.0 ppm or higher, then I'd take action by doing some water changes.
I hope that helps and good luck!
