Eq. III., the Lemniscate constant series, is proven using Virasoro Shapiro closed string amplitude Eq.(1.6) in arXiv:2409.06658v1 [math.CA] 10 Sep 2024 "String theory amplitudes and partial fractions" by HJALMAR ROSENGRENHJALMAR ROSENGREN, taking parameters $\lambda=s=0$ and $x_1=x_2=\frac14$, leaving out the series first term ($n=0$) and using Pochhammer $(a)_{-1}=\frac1{a-1}$.
Note that by Eq.(1.5) series terms are rational values even if Pochhammers have algebraic arguments.
Eq. II., Apery constant, this is the sketch of the proof. Assume that the Beta function has a generic expansion in terms of the sequence of functions $G_n(x,y)$ of the kind $$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}=\sum_{n=0}^\infty\,G_n(x,y)$$ differentiating and naming $\psi(x)$ the digamma function, $$B(x,y)\cdot\left[\psi(y)-\psi(x+y)\right]=\sum_{n=0}^\infty\frac{\partial}{\partial y}G_n(x,y)$$ By setting $y=1$ and $\gamma$ the Euler constant, we get $$\psi(x+1)+\gamma=-x\cdot\sum_{n=0}^\infty\frac{\partial}{\partial y}G_n(x,y)\Big|_{y=1}$$ Expanding the summands in Taylor series $$\frac{\partial}{\partial y}G_n(x,y)\Big|_{y=1}=\sum_{m=0}^\infty\,\frac{x^m}{m!}\cdot g_n^{(m)}$$ We get $$\psi(x+1)+\gamma=-x\cdot\sum_{m=0}^\infty\frac{x^m}{m!}\,\sum_{n=0}^\infty g_n^{(m)}$$ but $\psi(x+1)+\gamma$ is a GF for $\zeta(n)$ as $$\psi(x+1)+\gamma=-x\cdot\sum_{m=0}^\infty(-1)^{m+1}x^m\,\zeta(m+2)$$ So for $m\in \mathbb{Z}_{\ge0}$ $$\zeta(m+2)=\frac{(-1)^{m+1}}{m!}\cdot\sum_{n=0}^\infty\,g_n^{(m)}$$ The proven identity Eq.(1.3) in Rosengren's paper gives the summand sequence $G_n(x,y)$ by taking $x=x_1$, $y=x_2$ and the inert parameter $t=\lambda$. Apery's constant series II is obtained computing for $m=1$ the Taylor coefficients $g_n^{(1)}$. This is cumbersome but it is accomplished with the help of some CAS code. $$\zeta(3)=\sum_{n=0}^\infty\,g_n^{(1)}$$ where $g_n^{(1)}$ are the summands in Eq. II after shifting the sum index.
Eq. I. is proven in the @Hjalmar_Rosengren's answer below.