BG-11 medium works better in my experience. For marine species just add 3% NaCl. Do not bother with vitamins - simply gives contaminants a chance to grow. Diatoms? Make sure you add zero silicate and use plasticware not glass.
If you look through Stein and other algal culture books you quickly learn that most recipes have no rational basis. At best they are simply a recipe that worked and were never optimised or simplified. For example, the trace element mix f/2 added to seawater. The trace element mix is from Guillards freshwater trace element mix. It was siting on the shelf and was tried out and it worked or at least caused no harm. As an additive to seawater it is very unlikely that anything but the chelated iron has any significance.
Many recipes are wrong and the result of historical inertia. Like medieval herbals they are copies of copies of copies of copies. I have never been able to grow anything in Bold's Basal Medium as described (the trace metals are about 30-100 times too high) and Zarrouk's medium is another recipe that does not seem to ever work. The NPK fertilizer from the local supermarket (1g/l) is remarkably good for most purposes although you need to add some bicarbonate for BGA because the pH may be too low.
Yes! It does. The cyanobacteria is actually extreemophyles, that can grow/tolerate any extreem conditions. So, some species of marine cyanobacteria may like growing in G f/2 media.