Second-countable locally compact Hausdorff spaces are Polish, and homogeneous quotients are standard Borel

Let $\mathcal{B}$ be a countable base of $X$ and let ${\mathcal{B}}_{\mathrm{c}}$ be the members of $\mathcal{B}$ whose closure is compact. This is a base: given $x$ and an open $U\ni x$, [F1] yields an open $W$ with $x\in W\subseteq \bar{W}\subseteq U$ and $\bar{W}$ compact, and then some $B\in \mathcal{B}$ satisfies $x\in B\subseteq W$, so $\bar{B}\subseteq \bar{W}$ is compact and $B\in {\mathcal{B}}_{\mathrm{c}}$. Hence every point of $X$ lies in a member of ${\mathcal{B}}_{\mathrm{c}}$, whose closure is compact, and the closures of the countably many members of ${\mathcal{B}}_{\mathrm{c}}$ cover $X$; thus $X$ is a countable union of compact sets.

$X$ is regular by [F1] and $T_1$ because it is Hausdorff, so with its countable base it is metrizable by [F3]; fix a compatible metric $d$.

$X$ is Čech-complete by [F4], and being metrizable it is completely metrizable by the equivalence in [F4]; the choice hypotheses of [F4] are DC and the ultrafilter lemma with AC, which hold by [F2] under the standing AC. Since $X$ is second countable, [F5] makes it Polish, and then standard Borel by [F5]. This proves the first assertion.

Now let $G$ be a second-countable LCH group and $H\le G$ closed. By [F6] the quotient $G\mathrm{/}H$ is locally compact Hausdorff and $p$ is open, so the images $p(B)$ of the members of a countable base $B$ of $G$ form a countable family of open sets; it is a base of $G\mathrm{/}H$ because for $x\in G\mathrm{/}H$ and an open $U\ni x$ the preimage $p^{-1}(U)$ is open and contains a basic $B$ through some point of the fibre, whence $x\in p(B)\subseteq U$. Thus $G\mathrm{/}H$ is second-countable LCH, and [step 2.1] applied to $G\mathrm{/}H$ shows that $G\mathrm{/}H$ is Polish and its Borel structure is standard Borel.

The left action $a:G\times G\mathrm{/}H\to G\mathrm{/}H$, $a(g,xH)=gxH$, is continuous: the composite $(g,x)\mapsto p(gx)$ is continuous on $G\times G$ by [F7], it factors through the surjective open map $\mathrm{id}\times p:G\times G\to G\times G\mathrm{/}H$ (because $xH=x^{\mathrm{\prime }}H$ implies $gxH=g{x}^{\mathrm{\prime }}H$), and a continuous open surjection is a quotient map, so $a$ is continuous; in particular $a$ is Borel for the product of the Borel structures.

Combining the two parts: every second-countable LCH space is Polish and standard Borel, and for a second-countable LCH group $G$ with closed subgroup $H$ the homogeneous space $G\mathrm{/}H$ is Polish with standard Borel structure and the left action is continuous and hence Borel. The empty space is Polish and standard Borel by the same definitions, consistently with the vacuous case of the first assertion.