Ariyan,
EDIT: This contains some substantial edits and added references.
Lipman has defined the following notion (EDIT: twice):
Definition (Lipman ; Section 9 of "Rational singularities with applications to algebraic surfaces and factorization"): If $X$ is 2-dimensional and normal, $X$ has two pseudo-rational singularities if for every proper birational map $\pi : W \to X$ there exists a proper birational normal $Y$ over $W$ where, $R^1 \pi_* \mathcal{O}_Y = 0$
Definition (Lipman-Teissier ; Section 2 of "Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closures of ideals"): $X$ has pseudo-rational singularities if $X$ is CM (Cohen-Macaulay) and if for every proper birational map $\pi : Y \to X$ with $Y$ normal, $\pi_* \omega_Y = \omega_X$.
If these are the same in dimension 2, this seems pretty close to what you want in dimension 2.
EDIT2: These are the same in dimension 2, I was in Purdue and asked Lipman about question 1, which holds, and certainly implies this.
He also points out that regular schemes are pseudo-rational. In particular, this implies that if $\pi_* \omega_Y = \omega_X$ for one resolution of singularities, it also holds for every resolution of singularities (in fact, for every proper birational map with normal domain).
In dimension 2, he also studies relations between this condition and the local-finiteness of the divisor class group.
On the other hand, I'm pretty sure this is different from the definition of rational singularities you gave above at least in higher dimensions (with the appropriate $R^i$ vanishing instead of just $R^1$).
With regards to your specific questions:
Question #1: That vanishing, called Grauert-Riemenschneider vanishing, is known to fail for $\dim X > 2$ outside of equal characteristic zero. I believe the answer should hold in the two-dimensional case, certainly it should assuming that Lipman's various definitions of pseudo-rational singularities are consistent.
EDIT: This holds in dimension 2, see Theorem 2.4 in Lipman's "Desingularization of two-dimensional schemes".
In any dimension, that vanishing has recently been proven in equal characteristic $p > 0$ over a smooth variety (or a variety with tame quotient singularities), see arXiv:0911.3599.
Question #2: In higher dimensions, I'm pretty confident that the answer is no. In the 2-dimensional case, probably this is done by Lipman? In view of question #1, in order to find such a counter example in higher dimensions, one should look at various cones probably over 3 or 4-dimension schemes with negative Kodaira dimension (probably Fano's) but which fail Kodaira vanishing.
I have some thoughts on some other definitions of rational singularities which might be better in mixed characteristic, but I'm not sure I want to post them on MathOverflow right now. If you email me, I'd be willing discuss it a bit.
Quotient singularities can behave a little different outside of characteristic zero as well (see various papers of Mel Hochster from the 70s for instance). This can also lead one to look at questions like the Direct Summand Conjecture.
Best Answer
Now I can give you a definite answer. In general, the quotient of a klt singularity by a reductive group is not klt, because for instance, the canonical divisor of the quotient may not be $\mathbb{Q}$-Cartier.
However, one can define a broader notion: klt type. A singularity $(X;x)$ is said to be of klt type if there exists a boundary $B$ through $x$ for which $(X,B;x)$ is klt. Then, one gets the following theorem:
Theorem: Let $X$ be an affine variety with klt type singularities over an algebraically closed field of characteristic zero and $G$ be a reductive group acting on $X$. Then $X/\!/G$ is of klt type again.
The previous is Theorem 1 in https://arxiv.org/abs/2111.02812. I should also mention that the klt type property is an etale property, i.e., if you can check it in an etale cover of $X$ then it holds for $X$. This is Proposition 4.1. in https://arxiv.org/abs/2111.02812. Furthermore, the klt type condition