This is an exercise in 1.7. Spectral sequences of his Foundation of Algebraic Geometry
where $E^{\bullet,\bullet}$ is a double complex with $\mathrm{d}_{\to}:E^{p,q}\to E^{p+1,q}$ and $\mathrm{d}_\uparrow:E^{p,q}\to E^{p,q+1}$. The single complex $E^\bullet$ is $$E^k:=\bigoplus_{i\in\mathbb Z}E^{i,k-i},\quad \mathrm{d}^k:=\sum_i\big(\mathrm{d}_\to^{i,k-i}+\mathrm{d}_\uparrow^{i,k-i}\big). $$
I think the first statement that $H^0(E^\bullet)=E^{0,0}_\infty=E^{0,0}_2$ is wrong. Since $H^0(E^\bullet)$ in invariant under label-shift $(p,q)\rightsquigarrow(p-1,q+1)$, while $E^{0,0}_2$ is not. Am I correct?
Also, whether this can be rectified by adding some assumptions on $E^{\bullet,\bullet}$?
You're right. The exercise needs to assume that $E^{p,q} = 0$ if $p<0$ or $q<0$. This eliminates the invariance under reindexing.