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.
This is true and actually has nothing to do with the morphism. It's a simple fact about divisors and their associated reflexive sheaves.
So, $\omega_{X/C}$, the reflexive sheaf of rank $1$ associated to the Weil divisor $K_{X/C}$ is the reflexive hull of $\Omega_{X/C}^r$. In particular, there exists a natural morphism
$$
\Omega_{X/C}^r \to \omega_{X/C},
$$
which is neither necessarily injective nor surjective, but can be decomposed as a surjective morphism followed by an injective one.
$$
\Omega_{X/C}^r \twoheadrightarrow (\Omega_{X/C}^r)/({\rm torsion}) \hookrightarrow \omega_{X/C},
$$
It is easy to see that if you have a morphism
$$
\Omega_{X/C}^r \to \mathscr O(D),
$$
to a torsion-free sheaf, then it factors through $(\Omega_{X/C}^r)/({\rm torsion})$. On the other hand, since $\omega_{X/C}$ is the reflexive hull of $\Omega_{X/C}^r$ and hence also of $(\Omega_{X/C}^r)/({\rm torsion})$, any reflexive sheaf (e.g., a line bundle) that contains $(\Omega_{X/C}^r)/({\rm torsion})$, also contains $\omega_{X/C}$, which translated to divisors means that $K_{X/C}\leq D$.
This means that what you would like is actually true. In fact, $D$ doesn't even have to be Cartier, it works if it is Weil divisor.
Appendix (in response to the request in the comments):
As I mentioned, the morphism $\Omega_{X/C}^r \to \omega_{X/C}$ is simply the double dual. For any sheaf $\mathscr F$ you have a natural map
$\mathscr F\to \mathscr F^{\vee\vee}$. This is it.
Another way to think about it is to consider the embedding of the open set $\imath: U=X\setminus {\rm Sing}\, X\hookrightarrow X$ and then
$\mathscr F=\imath_*\imath^*\mathscr F$ so the above map is the natural map given by $1\to \imath_*\imath^*$.
For more on these issues, look at these MO answers:
S2 property and canonical sheaf.
Best Answer
Your hypothesis imply that $\omega_{Y/S}$ is an invertible sheaf (because $Y\to S$ is locally complete intersection).
(EDIT) As $f$ is flat at points of codimension $1$ ($Y$ is normal) and we are only interested on codimension 1 cycles, we can restrict $Y$ and suppose that $f$ is flat.
Then the dualizing sheaf $\omega_{X/Y}$ is invertible and you have the adjunction formula $$\omega_{X/S}=f^*\omega_{Y/S} \otimes\omega_{X/Y}.$$ The sheaf $\omega_{X/Y}$ is trivial outside of $B$ because $f$ is étale outside of $B$. It can be identified with the sheaf $\mathcal{Hom}_{O_Y}(f_{*}O_{X}, O_{Y})$.
Write $\omega_{X/Y}=O_X(D)$ for some Cartier divisor $D$ on $X$. Its support is contained in $f^{-1}(B)$. For any point $\eta$ of $X$ over a generic point $\xi$ of $B$, the stalk of $\omega_{X/Y}$ at $\eta$ is given by the different ideal of the extension of discrete valuation rings $O_{X,\eta}/O_{Y, \xi}$. The valuation of the different is known to be the ramification index $e_{\eta/\xi}$ minus $1$ when the ramification is tame and bigger or equal to $e_{\eta/\xi}$ otherwise (see Serre: Local fields). So the support of $D$ is equal to $f^{-1}(B)$ and is the ramification locus by definition.
In short, the coefficient of $R_f=D$ at the Zariski closure of $\eta$ is the valuation of the different ideal of $O_{X,\eta}/O_{Y, \xi}$. As for the computation, you can pass to the completions. A finite extension of complete DVR $R'/R$ is monogenous if the residue extension ($k(\eta)/k(\xi)$ in your case) is separable. If $R'=R[\theta]$, and $P(T)\in R[T]$ is the minimal polynomial of $\theta$, then the different ideal is generated by $P'(\theta)$. See Serre's book for more details.