BBA is generally the algae that results from too much light and not enough CO2, IME.
Very little eats it. Young true Siamese Algae Eaters are said to have a taste for it, but they lose it as they get older.
It's not harmful, just unattractive and can totally take over a tank. It's also one of the tougher algaes to deal with - tougher both physically and in terms of how hard it is to irradicate.
Many plants can handle a mild bleach dip for a short period of time. Thicker leafed plants like anubia resist the bleach very well. More delicate ones should not be bleached. Personally, I prefer to bleach only hardware and decorations and just prune infested leaves/plants. I've found that to be more successful since the plant remains healthy and keeps growing whereas after bleaching it seems to wilt and take some time to recover.