No, I would not classify B-Movie as a genre. The name came from the fact that it used to be the second movie of a double-bill, and was usually low budget and shorter than the other.
Nowadays we think of a B-Movie in a broader sense of being a low budget commercial movie, and has some overlap with Exploitation movies, but they can cover multiple genres including Horror as you mention. Science Fiction & Westerns are genres that have produced many B-Movies. The Italians were famous for producing 'Peplum' movies - also known as 'Swords and Sandals' movies.
It is true that there was a 'golden age' of B-Movies and that there are certain traits that might make you think 'this is a b-movie', but I do not think they would form a recognized genre in their own right.
Yes, director's would often know they were making a B-Movie - explicitly because it had a low budget and was destined to accompany a main feature. This might impart some of the characteristics that people love or hate about B-Movies, but it doesn't make it a genre IMHO.