For one, $\nabla \cdot \mathbf u \neq \mathbf u \cdot \nabla$:
The LHS is the divergence of $\mathbf u$, which is an expression, whereas the RHS is still an operator (in fact, $\mathbf u \cdot \nabla$ is called the advection operator, seen in the Navier-Stokes equations).
The issue here is that the commutative property of the dot product doesn't hold, because the dot product is supposed to be an operation between two vectors; $\nabla$ is an operator .
You're actually looking at an abuse of notation: you can interpret $\nabla \cdot \mathbf u$ intuitively, but need to be extra careful when performing algebraic manipulations.
The above also answers why the first term is not equal to the third term in your example; as for $\mathbf A(\nabla \cdot \mathbf B)$ and $(\mathbf A \cdot \nabla)\mathbf B$: the former is simply a scalar multiple of $\mathbf A$, whereas the latter is the result of some operation on the vector $\mathbf B$, which is much more complicated. Your impression that the two might be equal also involves "moving" the dot elsewhere, which can't be done either, even in the "usual" case.