This is all very helpful thank you so much!so I’ll try to answer, there are others who know more to these questions but I’ll try from my point of view:
everything using nitrate including coral and cyano, just need it
nitrate may rise a little if you choose to increase phos via feeding. I think the rise will be small or might even reduce if you increase phos significantly (which consistent roid feeding will do). Eventually both nitrate and phos should increase with “over feeding” your system.
I wouldn’t worry about the cyano. Siphon what you can out off the sand bed, make sure it’s not on any coral. Get your phosphate and nitrate at levels you want/coral like, eventually i bet that cyano dies out or moves to the sump. Microbial-ly I think the cyano is occupying a niche left by bacteria or others that aren’t getting their populations up bc of phosphate limitation. Cyano can use other sources of food (light, nitrate, etc) so will outcompete the starved bacs. Just give those bacteria the food they need and they should repopulate. Help them out by siphoning the cyano when you can.
don’t water change too much. It lowers nutrient (phos and nitrate). Water change after you get your tank “dirtier”