  We investigate the numerical approximation of integrals over $\mathbb{R}^d$
equipped with the standard Gaussian measure $\gamma$ for integrands belonging
to the Gaussian-weighted Sobolev spaces $W^\alpha_p(\mathbb{R}^d, \gamma)$ of
mixed smoothness $\alpha \in \mathbb{N}$ for $1 < p < \infty$. We prove the
asymptotic order of the convergence of optimal quadratures based on $n$
integration nodes and propose a novel method for constructing asymptotically
optimal quadratures. As for related problems, we establish by a similar
technique the asymptotic order of the linear, Kolmogorov and sampling
$n$-widths in the Gaussian-weighted space $L_q(\mathbb{R}^d, \gamma)$ of the
unit ball of $W^\alpha_p(\mathbb{R}^d, \gamma)$ for $1 \leq q < p < \infty$ and
$q=p=2$.
