$$\min_{P, Q} E_{x \sim P} -\log \frac{Q(x)}{P(x) + Q(x)} + E_{x \sim Q} -\log \frac{P(x)}{P(x) + Q(x)}$$
For the above problem, what are the minimizer $P$, $Q$? Can we say that it is minimized only for $P$, $Q$ satisfying $P(x) = Q(x)$ for any $x$? For P(x)=Q(x), the above term becomes $\log(4)$, is this the minimum possible value?
I expect your $P(x)$ and $Q(x)$ are densities representing distributions absolutely continuous with respect to some measure $\mathrm{d}x$. In that case, let $$D_f(P\parallel Q)\stackrel{\text{def}}{=}\int f\left(\frac{P(x)}{Q(x)}\right)Q(x)\,\mathrm{d}x$$ where $$f(z)\stackrel{\text{def}}{=}(1+z)\ln(1+z)-\ln z -(1+z)\ln 2\text{.}$$
$D_f(P\parallel Q)$ is $2\ln 2$ less than your functional.
Note that $f$ is strictly convex and $f(1)=f'(1)=0$. Consequently, $D_f(P\parallel Q)$ is an $f$-divergence, so
$$D_f(P\parallel Q)\geq 0\text{,}$$ with equality holding iff $P$ and $Q$ coincide as distributions.