  Using elliptic and parabolic regularity results in $L^p$-spaces and
generalized Dirichlet form theory, we construct for every starting point weak
solutions to SDEs in $\mathbb{R}^d$ up to their explosion times including the
following conditions. For arbitrary but fixed $p>d$ the diffusion coefficient
$A=(a_{ij})_{1\le i,j\le d}$ is locally uniformly strictly elliptic with
functions $a_{ij}\in H^{1,p}_{loc}(\mathbb{R}^d)$ and the drift coefficient
$\mathbf{G}=(g_1,\dots, g_d)$ consists of functions $g_i\in
L^p_{loc}(\mathbb{R}^d)$. The solution originates by construction from a Hunt
process with continuous sample paths on the one-point compactification of
$\mathbb{R}^d$ and the corresponding SDE is by a known local well-posedness
result pathwise unique up to an explosion time. Just under the given
assumptions we show irreducibility and the strong Feller property on
$L^{1}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$ of its transition function,
and the strong Feller property on
$L^{q}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$, $q=\frac{dp}{d+p}\in
(d/2,p/2)$, of its resolvent, which both include the classical strong Feller
property. We present moment inequalities and classical-like non-explosion
criteria for the solution which lead to pathwise uniqueness results up to
infinity under presumably optimal general non-explosion conditions. We further
present explicit conditions for recurrence and ergodicity, including existence
as well as uniqueness of invariant probability measures.
