Labour Productivity Growth and Reallocation of Labour
Ad 1)

You make a hand waiving claim. In these scenarios making concise mathematic statements helps improving your arguments.

Denote employment as $L$, define real value added as $F(L) - wL$, and productivity as real value / employment:

$$ A = \frac{F(L) - w(L)}{L}$$

The labor share should be some constant $\beta$ of total value added. Without going deep into mathematics, straight away assume $w(L) = \beta\alpha F(L)$. Hence, we have that

$$ A = \frac{F(L) (1 - \beta\alpha)}{L}$$

Now, what happens if $L$ increases? Let $L_2 &gt; L_1$

$$ A_2 - A_1 = (1-\alpha\beta)(\frac{F(L_2)}{L_2} - \frac{F(L_1)}{L_1})$$

Assume $F$ to be concave. Then we have that the change of productivity, $A_2 - A_1$ is negative - as you say.

However, $A$ here is average productivity. Total productivity is $AL$.

$$ A_2L_2 - A_1L_1 = (1-\alpha\beta)(F(L_2) - F(L_1))$$.

Independent of the curvature of $F$, for any strictly increasing production function, this will be positive.

tl;dr 


Don't confuse average productivity with total productivity. 
Whenever you find something weird, use algebra to make your statements precise


Ad 2)

Again, you are not being precise enough. What is the government's objective? If government is only about maximizing average productivity, the solution is to let $L\to 0$, as then $F'(L) \to \infty$ (and for Cobb-Douglas, $F'(L)$ is proportional to average productivity).

Maximizing total output implies allocating workers to the sectors with highest productivity. I do not understand how the functional relationship between employment intensity and labor productivity affects that premise. Furthermore, unless there is a market distortion, the role of any government intervention is not clear at all.