Yes to the first question. By shift I mean that you should pay attention to the degrees of the complex, but since you have understood the restriction condition you are fine. Just read on (and read examples) and everything will make sense.

This can't be true, as the determinant of the image contains only $k-1 $-th powers (there are other much more subtle obstructions).

@yhb If $\varphi\in E$, $T_p\circ W_N\varphi(f)=T_p\varphi(f^{\sigma})=a^{\sigma}_p\varphi(f^{\sigma})$ and the only quotient of $J_1$ satisfying this is $\bar{E}$.