= Solution
We use the following standard <planar Brownian motion> facts: its exit <probability distribution> is <harmonic measure>; <conformal maps> preserve that <probability distribution> after an increasing time change; and the <Poisson kernel for the upper half-plane> gives its real-boundary exit density. For rough hulls the <boundary> is interpreted through <prime ends>, with <boundary> correspondence at harmonic-measure-almost every exit point. This avoids imposing a smoothness or local-connectedness assumption on $K$.
Under $g_K$, the exit event through $K$ corresponds to a measurable set $E_K\subset\mathbb R$ of <boundary> images. It is bounded: the <mapping-out function> extends by <Schwarz reflection> along the real line outside a sufficiently large interval enclosing $K$, and its <boundary> near infinity corresponds to the original real <boundary> near infinity. All <prime ends> representing $K$ therefore have images in a bounded interval. The <boundary> correspondence assertion can equivalently be formulated using the almost-everywhere limits of $g_K^{-1}$; changing null sets does not change $E_K$'s measure.
Write $g_K(iy)=u_y+iv_y$. The <hydrodynamic normalization at infinity> gives $u_y=O(y^{-2})$ and $v_y=y-a_K/y+O(y^{-2})$. By <conformal invariance of planar Brownian motion> and the <Poisson kernel for the upper half-plane>,
$$
\mathbb P_{iy}(B_T\in K)=\frac1\pi\int_{E_K}\frac{v_y}{(s-u_y)^2+v_y^2}\,ds.
$$
On the bounded set $E_K$, the integrand multiplied by $y$ tends uniformly to one. Therefore the limit exists and is finite:
$$
\boxed{c(K)=\frac{|E_K|}{\pi}.}
$$
This is the <harmonic capacity from infinity in the upper half-plane> with the normalization $c(K)=\operatorname{cap}(K)/\pi$; it is different from <half-plane capacity>. If real <boundary> points are retained in the definition of $K$, include their <boundary> images in $E_K$ as well; the argument is unchanged.
For the scaling statement, <Brownian scaling> says that $r^{-1}B_{r^2t}$ started from $iy/r$ is again standard <planar Brownian motion>. The exit event scales with the hull, so
$$
\mathbb P_{iy}(B_T\in rK)
=\mathbb P_{i(y/r)}(B_T\in K).
$$
Set $v=y/r$, multiply by $y=rv$ and take the limit. It follows that
$$
\boxed{c(rK)=r\,c(K).}
$$
The same conclusion follows geometrically from $E_{rK}=rE_K$ and the formula above.
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