Say $X=\mathbb A^n$ and $D_1,\dots, D_n$, the components of $D$, are the coordinate hyperplanes $x_i=0$, for simplicity. You can assume that WLOG, because your $(X,D)$ is isomorphic to this one in étale topology.
$\pi_1(X\setminus D) = \mathbb Z^n$. So the cover $V\to U$ corresponds to a finite quotient of $\mathbb Z^n$, which is a finite abelian group $G$. So the ring of regular functions on $V$ is generated by the roots of monomials in $x$. You can write these as $x^m$ for some $m\in \mathbb Q^n$. The lattice $H$ generated by $m_i$ contains $\mathbb Z^n$, and the quotient $H/\mathbb Z^n$ is the dual abelian group $G^{\vee}$.
So what is $Y$ now? It is the normalization of the ring $k[x_1,\dots,x_n]$ in the bigger field $k(x^{m_i})$. So it is a toric problem now. The normalization is generated by monomials in the lattice $H$ which lie in the cone $(\mathbb R_{\ge0})^n$.
So $Y$ is toric and simplicial, and every such singularity is an abelian quotient singularity.
The condition $\pi_1(X\setminus D) = \mathbb Z^n$ fails in char $p$ (indeed, the fundamental group in that case is huge), so this argument and the statement both fail in char $p$.
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
What embedded resolution does achieve is this: if $Z\subset X$ is a Zariski closed set in a variety, then there is a smooth variety $Y\to X$, obtained from a series of blow ups along smooth centers, such that the preimage of $Z$ is a divisor with normal crossings. However, there is no reason to expect that the inverse of image of $Z$ (or your $D$) in a further blow up will be smooth. (Francesco said this already in a comment, but it bears repeating.) This is already clear in the case where D is a union of two lines in the plane. If you take the preimage in a further blow up, you will get a tree of $\mathbb{P}^1$'s. So your instinct is correct here.