Regarding question 1), your septic has discriminant $-7\cdot f(h)^2$ (for a suitable polynomial $f(h)$, so the quadratic subextension of the splitting field over $\mathbb{Q}(h)$ is $\mathbb{Q}(h)(\sqrt{-7})\subset\mathbb{Q}(h)(\zeta_7)$. Making the expression $y=u_1^{1/7}+u_2^{1/7}+u_3^{1/7}$ well-defined requires picking the correct 7-th roots inside the splitting fields $\mathbb{Q}(\sqrt[7]{u_i}, \zeta_7)$, so I guess this is where the above quadratic subextension gets eaten up.

Regarding question 1), of course the obvious (sufficient) answer is "When the Galois group is contained in $C_7\rtimes C_3$". That's not quite the case here, but "almost". To be precise, your septic has discriminant $-7\cdot f(h)^2$ (for a suitable polynomial $f(h)$, so the quadratic subextension of the splitting field over $\mathbb{Q}(h)$ is $\mathbb{Q}(h)(\sqrt{-7})\subset\mathbb{Q}(h)(\zeta_7)$. Making the expression $y=u_1^{1/7}+u_2^{1/7}+u_3^{1/7}$ well-defined requires picking the correct 7-th roots inside the splitting fields $\mathbb{Q}(\sqrt[7]{u_i}, \zeta_7)$, so I guess this is where the above quadratic subextension gets eaten up.