The answer to the first question, is $X\circ Y$ a vector field if $X$ and $Y$ are, is in general no. The reason is that the operator does not satisfy the Leibniz rule. One of the cool things about the Lie bracket of vector fields is that the problematic terms vanish and it is a vector field.
The question about Baker-Campbell Hausdorff formula is not well stated, you could make it true by being careful about what you mean or make it false.
Finally, $B(X,Y)$ makes sense: you are evaluating a smooth two form on two smooth vector fields.
My guess is that you really want to be asking something about a deformation quantization of a subspace of the smooth functions on the manifold.
