A linear uniformly elliptic equation of the form $F(D^2u) = 0$ can only be $\Delta u = 0$ (up to an affine transformation of the solution) since the only inputs are the second derivatives. Equations like $a^{ij}(x)u_{ij} = 0$ are of the form $F(D^2u,x) = 0$. If $F(.,x_0)$ is concave for each $x_0$ (corresponding to freezing the coefficient in the linear setting) and $F$ is Holder continuous in $x$, then we get $C^{2,\alpha}$ regularity by applying Evans-Krylov and a perturbation argument (see ch. 8 of Caffarelli-Cabre). This is a generalization of Schauder estimates.

A more interesting concave equation "built out of" linear ones is the Bellman equation,
$$\inf_{\alpha}a^{ij}_{\alpha}u_{ij} = 0$$
where $a^{ij}_{\alpha}$ is a collection of uniformly elliptic constant-coeffient matrices.

For the second question, the proof actually gives $C^{2,\alpha}$ regularity for a $C^{1,1}$ viscosity solution in all of $B_{1/2}$. Indeed, one shows the oscillation decay of $D^2u$ (as an $L^{\infty}$ map)
$$diam(D^2u(B_{\delta^k}(x))) < 2^{-k}\|u\|_{C^{1,1}(B_1)}$$
for some universal $\delta$. It follows that $D^2u$ is Holder continuous up to modification on a set of measure $0$. By using quadratic approximations to $u$ at nearby points it is straightforward to show that $u$ is in fact $C^{2,\alpha}$ everywhere. (Think for example that the function $\chi_{\{0\}}$ cannot be the derivative of any differentiable function, since if it were it would lie close to a line of slope $1$ near $0$ and violate being locally constant away from 0).

As a side note, it is an interesting result of [Armstrong, Silvestre and Smart][1] that for any uniformly elliptic equation $F(D^2u) = 0$ with $F \in C^1$ we get $C^{2,\alpha}$ regularity off of a set of Hausdorff dimension $n-\epsilon$ for some small universal $\epsilon$.


  [1]: http://arxiv.org/abs/1103.3677 "Armstrong, Silvestre and Smart"