Sum of random variables of strictly stationary series If we know that the distribution of $(X_t, X_{t+1}, \dots, X_{t+h})$ and the distribution of $(Y_t, Y_{t+1}, \dots, Y_{t+h})$ are independent of $t$, for every $h \in \mathbb{N}$, does this imply that the distribution of $(X_t + Y_t, \dots, X_{t+h}+ Y_{t+h})$ is also independent from $t$?

Not necessarily. Consider $(X_t)$ i.i.d. with a symmetric distribution and $Y_t=(-1)^tX_t$. Then $(X_t)$ and $(Y_t)$ are stationary but $X_t+Y_t$ is alternatively $0$ and $2X_t$ hence $(X_t+Y_t)$ is not stationary.
