The Biozyme will only do what your nitrifying bacteria are already doing--oxidizing ammonia. That is a waste of money, once your tank is cycled and you have these cultures established. The end result of the nitrification cycle will always be nitrates.
The best way to make a significant dent in the nitrates is to begin doing daily water changes of 50% or better. Also, consider adding live rock to equal 1-2lbs/gallon, and make it very good quality live rock. And, I will suggest considering adding a refugium OR deep sand bed (DSB) directly to the tank. Refugia methodology makes use of the nutrient exporting capabilities of macroalgae, and when used in conjunction with a DSB it can really reduce available nutrients (you would have to directly export nutrients from the refugium via harvest of macroalgae).
As for the hair algae, consider something like a tuxedo urchin for direct, immediate consumption.