Understanding the definition of a "restriction of a resolution derivation"
For the first point, it seems that $C_{j_1} := D_{j_1} \lor y$ and $C_{j_2} := D_{j_2} \lor \bar{y}$ are meant to be the clauses in $\pi$, instead of $C_{j_1} \lor y$ and $C_{j_2} \lor \bar{y}$. This may help in understanding such a definition. Furthermore, it seems that the resolution of $C'_{j_1}$ and $C'_{j_2}$ upon $y$ is taken, rather than $C'_{j_1} \lor C'_{j_2}$ (they are equivalent).
This view would explain, for example, the computation of $C_4[x=0]$: $C_1' = x_2 \lor \bar{x_3}$ and $C_2' = x_3$ are resolved upon $x_3$ to obtain $C_4' = x_2$.

For the second point: the computations are meant to produce the same output. In the paper, the final result without any computation is shown. Going through the recursive computation by pencil and paper shows how the rules are used.