1.) Nitrogen-fixation, denitrification, and nitrification are three different processes.
Nitrogen-fixation is a process wherein soil bacteria convert atmospheric nitrogen into nitrate, which becomes available to plants through the roots. Other soil bacteria called denitrifiers can do the opposite, converting nitrate in the soil into nitrogen which is then lost to the atmosphere. This process occurs when there is poor oxygenation of the soil.
These processes are part of the over-all nitrogen cycle in nature, but are not significant players in what the aquarium hobby calls "cycling". This usually refers to the nitrification process wherein ammonia is converted to nitrite then to nitrate. In healthy well-oxygenated aquariums, this is a one-way process, and the nitrate will eventually concentrate in the system. Plants in aquariums require nitrogen and can use nitrate, thus completing the full cycle. Denitrification is a subject that can be of interest to planted aquarium enthusiasts, but not so much nitrogen-fixation. Aquatic plants have no known means of using atmospheric nitrogen directly.
Cyanobacteria are photosynthesizing bacteria, sometimes called blue-green algae. Various species occur in the aquarium under certain circumstances. They don't play an especially significant role in any of these processes.
2.) Dr Timothy Hovanec has shown that the nitrifiers responsible for conversion of nitrite to nitrate in the aquarium are mostly in genus Nitrospira, rather than in genus Nitrobacter, a common nitrifier in soils. Dr Tim and colleagues have done research into the microbiology of organisms that are important to biological filtration.