Confusion regarding Riemann-Roch for vector bundles

Let $k$ be an infinite non-algebraically closed field, $X$ a smooth projective curve on $k$ and $E$ a locally-free sheaf on $X$ of rank at least $2$. Denote by $\bar{k}$ the algebraic closure of $k$, $X_{\bar{k}}$ the base-change of $X$ to $\bar{k}$ and $E_{\bar{k}}$ the pull-back of $E$ to $X_{\bar{k}}$.

Is the Riemann-Roch formula for $E$ the same as for $E_{\bar{k}}$? In particular, if $E_{\bar{k}}$ has a non-trivial global section then does it necessarily imply that $E$ too has a non-trivial global section defined over $k$?

Riemann--Roch is "the same" over any field. But in any event, this isn't so helpful for you since it only shows that the Euler characteristic $h^0(X,E)-h^1(X,E)$ is the same over $\bar k$ as it is for $k$.
The key phrase in your setting is "cohomology and base change". Cohomology and base change results tell you when taking cohomology commutes with base change. The "base change" in question is $\operatorname{Spec}\bar k\to\operatorname{Spec}k$, so "cohomology commuting with base change" is the statement that the natural map $H^\bullet(X,E)\otimes_k\bar k\to H^\bullet(X_{\bar k},E_{\bar k})$ is an isomorphism. In your case, this holds since your base change $\operatorname{Spec}\bar k\to\operatorname{Spec}k$ is flat, so you can apply Lemma 29.5.2 (Flat base change) from http://stacks.math.columbia.edu/tag/02KE.