Also, how much does it "capture" carbon as part of the photosynthesis ?
https://www-cyanosite.bio.purdue.edu/media/table/media.html
Mostly they need nitrogen, phosphate, and sulfur to make protein and nucleic acids, and trace metals,ions, and some vitamins to use as cofactors for enzymes. In the wild, the limiting nutrient for cyanobacteria is often iron or nitrogen.
They don't need an added carbon source in their media since they get to eat dissolved CO2 from the air. Every carbon atom in newly synthesized molecules comes from CO2, and there are (very roughly) 10^10 carbon atoms per bacterial cell. http://book.bionumbers.org/what-is-the-elemental-composition... . So if you know the growth rate you can estimate the carbon fixation rate.
How can nitrogen be a limiting factor if ~78% of air is nitrogen? I'm assuming they also take the CO2 out of the air, which just makes up 0.04%.
Taking up inorganic nitrogen is called nitrogen fixation and takes a bunch of machinery and energy. Also, oxygen inhibits the process, so for photosynthetic bacteria a bunch of extra steps have to be taken to compartmentalize photosynthesis and nitrogen fixation. So not all cyanobacteria can do it, and even the ones who are able to will prefer to take up organic nitrogen if available, and they grow slower when they have to use N2.
Nitrogen in air is not normally available to organisms. The triple bond between the 2 nitrogen atoms is very strong and takes a good deal of energy to break. Some bacteria can do this, and legumes famously have evolved to house bacteria capable of fixing nitrogen in their roots.
The main way we get nitrogen to agricultural plants today is by the industrial production of ammonia, which takes enormous amounts of energy and lots of fossil fuels as presently implemented.
Carbon is captured through photosynthesis and is stored as biomass. But when the microalgae dies the carbon is released again. So to truely store carbon the microalgae needs to be cultivated and buried / contained.