First Order Stochastic Dominance given a functional form Assume I have a particular function $u(x)= \sqrt x $ (note: $u'(x)\geq0,  u'(x) \leq 0, \forall x$)
Assume that I have two continuous, probability distributions over $x$, $F(x)$ and $G(x)$ but that they are unknown (we know they exist but we dont know what they look like)
Assume the following about the expectations given the two distributions: $\int u(x)dF(x) \geq \int u(x)dG(x)$
(1)
Given the information above, based on one specific functional form of $u(x)$, can I conclude: $F(x)\leq G(x), \forall x$?
(2)
If I assume $u(x)$ such that $\quad u'(x)\geq0, \quad u'(x) \leq 0 \quad \forall x$
(but do not specify what functional form $u(x)$ takes)
and I know $\int u(x)dF(x) \geq \int u(x)dG(x)$
can I conclude: $F(x)\leq G(x), \forall x$ ?
 A: For (1), the answer is clearly no. For example, let $F(x)$ be the uniform distribution over $[0,1]$ and $G(x)$ be the cdf of the degenerate distribution
$$\mathbf P(X=1/2)=1.$$
Then it is easy to see that 
$$F(x)=\begin{cases}0, &\text{if }x<0\\x,&\text{if }0\le x<1\\1,&\text{if } x\ge 1\end{cases},\,\,\,G(x)=\begin{cases}0,&\text{if }x<1/2\\1,&\text{if }x\ge 1/2\end{cases}.$$
Clearly, (Note that although $G$ is not continuous here, it can be arbitrarily approximated by a distribution function that is continuous)
$$\int_{-\infty}^\infty u(x)\,dF(x)=\frac 23< \int_{-\infty}^\infty u(x)\,dG(x)=\sqrt{\frac 12}.$$
But clearly, neither "$F(x)\ge G(x)$ for all $x$" nor "$G(x)\ge F(x)$ for all $x$" holds.
For (2), if you mean "for SOME $u(x)$ satisfying $u'\ge 0$ and $u''\le 0$, the inequality holds", then the answer is clearly the same as (1), since $u(x)$ could rightly be the function you specified in (1). But if you mean "for ALL $u(x)$ satisfying $u'\ge 0$ and $u''\le 0$, the inequality holds", then the answer is yes, because $F$ first-order stochastically dominates $G$ if and only if 
$$\int u(x)\,dF(x)\ge\int u(x)\,dG(x)$$
for EACH concave and increasing function $u(x)$, while each concave and increasing function can be approximated by a function which is twice differentiable, increasing, and concave.
