I think Q1 has a positive answer. To see this, review how the results are proved in DFJP. You have a weakly compact subset $W$ of $Y$ (the closure of the image of the unit ball of $X$ under a weakly compact operator) and you apply the factorization technique (which we now know is just a real interpolation method applied to the pair $(W, B_Y)$) to $W$ to obtain a larger weakly compact symmetric subset $C$ of $Y$. Consider the normed space that has $C$ as its unit ball. It is proved that $C$ is weakly compact in in this normed space and hence this normed space is reflexive.

When $Y$ has a (always Schauder) basis, instead of applying the interpolation technique to the original weakly compact set $W$, we enlarged $W$ to a bigger weakly compact set $W_1$ that is invariant for the partial sum projections associated with the basis for $Y$. Then the interpolation technique, applied to $W_1$, produces a still larger weakly compact set $C$ that is invariant for the partial sum projections associated with the basis for $Y$. From this it is easy to see that the basis for $Y$ is also a basis for the space that has $C$ as its unit ball.

For $W_1$ we used the closed convex hull of the union over $n$ of $P_n W$, where $(P_n)$ are the partial sum projections for the basis for $Y$. When the basis is shrinking, we proved that $W_1$ is weakly compact. Suppose that $Y=L_1$ and you use the Haar basis for $Y$. Even though the Haar basis is far from being shrinking, the same construction produces a weakly compact set $W_1$. To see this, recall that a weakly closed set $D$ in $L_1$ is weakly compact iff for all $\epsilon > 0$ there is $M=M_\epsilon$ s.t. $D\subset M B_{L_2} + \epsilon B_{L_1}$ (``uniform integrability") and use the fact that both $B_{L_2}$ and $B_{L_1}$ are invariant under the Haar basis projections.