If $L_1$ is a sublattice of $L_2$, it does not mean that the Hasse diagram of $L_1$ is a subgraph of the Hasse diagram of $L_2$. The order in a lattice is transitive, so you can connect two elements in a sublattice that were not connected in the lattice. I believe you can take $c$ or $e$ as the rightmost element of $L_1$.

Concerning (ii), according to Wikipedia, a sublattice is not a subset that is a lattice, but a subset that is closed under the original meet and join operations. Since there are two shaded elements whose join in the sense of $L_3$ is not shaded, the shaded elements don't form a sublattice.