I believe it depends on the strain, but certainly the model species Synechocystis sp. PCC 6803 can survive and grow just fine in anoxic conditions with seemingly no effect on photosynthesis. Of course, if they are photosynthesising they are making their own oxygen so it could be a most a low-oxygen environment.

Regarding nitrogen fixation, the answer is yes - again depending on the species. Many heterocystous species (e.g. Nostoc punctiforme) can perform nitrogen fixation, and many single-celled forms such as Cyanothece can also do this.

Dr Tim Crawford thank you for your contribution to the discussion

Any organism which can exist or live without oxygen is called an anaerobic organism or anaerobes. Anaerobic bacteria are also examples of anaerobic organisms, which do not require oxygen for respiration. Blue-green algae are also known as Cyanobacterium. Many species of cyanobacteria not only fix carbon in CO2 through photosynthesis, but they can also fix atmospheric nitrogen. Both of these processes also play an important role in humus formation. Many species of cyanobacteria not only fix carbon in CO2 through photosynthesis, but they can also fix atmospheric nitrogen. Both of these processes also play an important role in humus formation. Cyanobacteria are oxygenic photosynthetic bacteria that are widespread in marine, freshwater and terrestrial environments, and many of them are capable of fixing atmospheric nitrogen. Cyanobacteria such as Nostoc and Anabaena help in nitrogen fixation. They form specialized cells called 'heterocysts'. Heterocysts contain nitrogenase enzyme that converts nitrogen to ammonia. They live in water and can manufacture their food through the process of photosynthesis and releasing oxygen. They can fix atmospheric nitrogen as free-living forms and in mutual association with the roots of plants. Other than nitrogen cyanobacteria can also fix carbon from carbon dioxide during photosynthesis.