NO3 testing near zero is not a problem. We do not have accurate test kits to measure it and you are not truly at zero or else stuff would be dead and there would be no algae grow. For N and P, testing for an actual number and then chasing it is fools gold combined with a fools errand... throughput is more important. There are a lot of really good tanks, and the ocean, that test near zero, but they have a good amount of import and export so that there is always some around. However, not many of these tanks use GFO and organic carbon since that can drive either too low. The bottom line is that near-zero is more than OK, but zero is usually bad.
Chaeto will not starve out the hair algae - they can live in harmony if you let them. The idea is that you remove, or get consumers, for the hair algae to eradicate it and the chaeto continues to grow in an area with no predators. If you do not regularly change water, chaeto will use up all of the iron in the system and then stop growing. Water changes will add it back, but if you are not doing them, then you might need to dose iron.
I would make sure that the whites are down pretty low, and move the monti down in the tank. That is not a great light to light SPS that are not in ideal water situations and do not seem capable of "picking up" other deficiencies in the system. While you see some nice tanks with SB Reef Lights, they seem to rarely have other issues to deal with. Once you get everything figured out and nice and stable, then you can probably turn the lights back up some.
