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
I believe the answer is yes, although I do not know of a reference.
We will use the original definition of pseudo-rationality due to Lipman and Teissier [Lipman–Teissier 1981, p. 102], and the following characterization:
Lemma [Lipman–Teissier 1981, Remark (a) on p. 102]. Let $(R,\mathfrak{m})$ be an $n$-dimensional normal Cohen–Macaulay local ring. Then, $R$ is pseudo-rational if and only if for every projective birational morphism $f\colon W \to \operatorname{Spec}(R)$ where $W$ is integral, there exists a proper birational morphism $g\colon W' \to W$ such that $W'$ is normal and integral and such that $$\delta_{fg}\colon H^n_{\mathfrak{m}}(R) \longrightarrow H^n_{(fg)^{-1}(\{\mathfrak{m}\})}(\mathcal{O}_{W'})$$ is injective.
Proposition. Let $(R,\mathfrak{m})$ be an $n$-dimensional noetherian local ring that is pseudo-rational, and let $\varphi\colon R \to S$ be an étale map. Then, for every prime ideal $\mathfrak{n} \subseteq S$ lying over $\mathfrak{m}$, the ring $S_{\mathfrak{n}}$ is pseudo-rational.
Proof. Since $\varphi$ is étale, $S$ is normal and Cohen–Macaulay [Matsumura 1989, Corollary to Theorem 23.9 and Corollary to Theorem 23.3], and $S_\mathfrak{n}$ is of dimension $n$ [Stacks, Tag 04N4].
Now let $f_\mathfrak{n}\colon W_\mathfrak{n} \to \operatorname{Spec}(S_\mathfrak{n})$ be a projective birational morphism, where $W_\mathfrak{n}$ is integral. Since $S_{\mathfrak{n}}$ is integral and noetherian, there exists an ideal $I_{\mathfrak{n}} \subseteq S_{\mathfrak{n}}$ such that $W_\mathfrak{n}$ is the blowup along $I_{\mathfrak{n}}$ [EGAIII$_1$, Corollaire 2.3.6]. Clearing denominators, there exists an ideal $I \subseteq S$ localizing to $I_{\mathfrak{n}}$ such that the blowup $f\colon W \to \operatorname{Spec}(S)$ along $I$ localizes to $f_\mathfrak{n}$.
By [Stacks, Tag 087B] or [Rydh, Proposition 4.14], there exists an ideal $J \subseteq R$ such that we have the commutative diagram $$\require{AMScd}\begin{CD} W' @>g>> W @>f>> \operatorname{Spec}(S)\\ @VVV @. @VVV\\ \operatorname{Bl}_J\bigl(\operatorname{Spec}(R)\bigr) @= \operatorname{Bl}_J\bigl(\operatorname{Spec}(R)\bigr) @>h>> \operatorname{Spec}(R) \end{CD}$$ where $W' \to \operatorname{Spec}(S)$ is the blowup along $JS$, and the outer rectangle is cartesian by flat base change [Stacks, Tag 0805]. Replacing $\operatorname{Bl}_J(\operatorname{Spec}(R))$ by its normalization, we may assume that $\operatorname{Bl}_J(\operatorname{Spec}(R))$ is normal.
Now by flat base change for local cohomology [Hashimoto–Ohtani 2008, Theorem 6.10], the homomorphism $$\delta_{f_\mathfrak{n}g_\mathfrak{n}}\colon H^n_{\mathfrak{n}}(S_{\mathfrak{n}}) \longrightarrow H^n_{(f_\mathfrak{n}g_\mathfrak{n})^{-1}(\{\mathfrak{n}\})}(\mathcal{O}_{W'_{\mathfrak{n}}})$$ is obtained from $\delta_h$ by tensoring with $S_{\mathfrak{n}}$, and hence is injective. Finally, since $fg$ is a blowup and $W'$ is normal by [Matsumura 1989, Corollary to Theorem 23.9], we see that $S_\mathfrak{n}$ is pseudo-rational by the Lemma above. $\blacksquare$
It should be possible to prove that completions of pseudo-rational G-rings are pseudo-rational, using Néron–Popescu desingularization [Popescu 1986, Theorem 2.4; Popescu 1990, p. 45; Swan 1998, Theorem 1.1] to reduce to showing that pseudo-rationality is preserved under étale extensions (shown above) and polynomial extensions.