Production under uncertainty
Under random prices, the producer solves

$$\max_x E[U(G(x,p)]$$

Compacting $U(G(x,p) \equiv U$, the f.o.c is

$$E[U'\cdot (p_x-2cx)]= 0 \implies x^* = \frac 1 {2c}\cdot \frac {E[U'\cdot p_x]}{E[U']} \tag{1}$$

or

$$x^* = \frac 1 {2c}\cdot \left[E(p_x) + \frac {\text{Cov}(U', p_x]}{E[U']}\right] \tag{2}$$

Now, 
$$U' = \frac 1{2(p_xx - cx^2)^{1/2}}$$ 

and $U'$ is decreasing in $p_x$. Then (see for example Egozcue, Cogent Mathematics (2015), 2: 991082), Theorem 2(2) p. 6)

$$\text{Cov}(U', p_x] &lt;0$$

So we conclude that 

$$\implies x^* &lt; \frac {E(p_x)} {2c}$$

the RHS being the risk-neutral solution.

Plugging the distributional assumption for $p_x$ in $(1)$ gives us an implicit function for $x^*$. If one is patient enough to go through some boring algebra after that, one will obtain a very simple nice looking closed-form solution. 

As for the last question, it is poorly worded and it does not make much sense, or it is a trick question, since the set up implies that the producer must commit to an output level before the price is actually realized. If this was not the case, then the producer would have no reason to solve an expected utility maximization problem, but he would wait to see the actual price and optimize accordingly. 

If the last question alludes to a dynamic maximization problem, then this is a whole new situation, and additional assumptions and modelling are needed in order to proceed.