It is not true that $\text{rank}_+^*(V) \leq \text{rank}_+(V) $.  In fact, an equivalent definition of the non-negative rank of $V$ is the minimum number of non-negative vectors (not necessarily columns of $V$) such that every column of $V$ is a conic combination of these vectors.  Therefore, the opposite inequality  $\text{rank}_+^*(V) \geq \text{rank}_+(V) $ holds.  

For an explicit example where $\text{rank}_+^*(V) > \text{rank}_+(V) $, consider the matrix 

$$
V=
  \begin{bmatrix}
    1 & 0 & 1 & 2  \\
    0 & 1 & 2 & 1  \\
    0 & 0 & 1 & 1 
  \end{bmatrix}
$$

It is easy to see that $\text{rank}_+^*(V) =4 $ and $\text{rank}_+(V)=3$. This example also shows that your claimed inequality $\text{rank}_+^*(V) \leq n-1 $ does not always hold.