EDIT: The OP added some questions, including one about the case of a right half-plane cohomological bicomplex $(B^{i,j}, d_h, d_v)$, where $B^{i,j} = 0$ for $i<0$, $d_h : B^{i,j} \to B^{i+1,j}$ and $d_v : B^{i,j} \to B^{i,j+1}$. Suppose that $Z^{i,j} \subset B^{i,j}$ is such that $d_h(Z^{i,j}) = 0$, and filter the total complex $$ (C, d) = (\bigoplus_{i,j} B^{i,j}, d_h + d_v) $$ by $$ F^s = \bigoplus_{j-i>s} B^{i,j} \oplus \bigoplus_{j-i=s} Z^{i,j} $$ for all integers $s$. Here $C$ and $F^s$ are graded, with $B^{i,j}$ and $Z^{i,j}$ in degree $i+j$. Then $(F^s, d)$ is a subcomplex of $(C, d)$, and contains $(F^{s+1}, d)$ as a further subcomplex. We get an exact couple in the usual way, with $A^{s,t}$ and $E_1^{s,t}$ equal to the degree $s+t$ parts of $H(F^s, d)$ and $H(F^s/F^{s+1}, d)$, respectively.

I claim that $\lim_s F^s = 0$ and $\lim^1_s F^s = 0$. This can be checked one degree at a time, since $F^s = 0$ in degrees less than $s$. It follows (see Boardman's Theorem 9.2) that $\lim_s A^s = 0$ and $\lim^1_s A^s = 0$, so the spectral sequence is conditionally convergent to the colimit $G = H(C, d)$.

Furthermore, $F^s/F^{s+1}$ and $E^1_s$ are concentrated in degrees $\ge s$, corresponding to $t\ge0$, so this is an upper half-plane cohomological spectral sequence with exciting differentials. Hence it is strongly convergent to $G$, by the modified form of Boardman's Theorem 6.1 that I mentioned above.

To prove the modified form, one does as Boardman says. Let $F^s G$ be the image of $H(F^s, d)$ in $G$. One must check that the filtration $\{F^s G\}_s$ of $G$ is complete Hausdorff and exhaustive, and that the natural inclusion $F^s G/F^{s+1} G \to E_\infty^s$ is an isomorphism. Both claims can be checked one degree at a time, and for each degree the proof of Theorem 6.1(a) carries over. (I do not have Cartan--Eilenberg at hand: I do not recall if they spelled this out.)

After a long life in preprint form, Boardman's paper was published in the conference proceedings celebrating his 60th birthday:

  \bib{MR1718076}{article}{
       author={Boardman, J. Michael},
       title={Conditionally convergent spectral sequences},
       conference={
          title={Homotopy invariant algebraic structures},
          address={Baltimore, MD},
          date={1998},
       },
       book={
          series={Contemp. Math.},
          volume={239},
          publisher={Amer. Math. Soc., Providence, RI},
       },
       date={1999},
       pages={49--84},
       review={\MR{1718076}},
       doi={10.1090/conm/239/03597},
    }

In the $(s,d)$-bigraded case, you can replace Boardman's (left half-plane) condition that $E_1^{s,d} = 0$ for $s > 0$ (or $s > s_0$ for some fixed integer $s_0$) by the (upper half-plane) condition that $E_1^{s,d} = 0$ for all $d < 0$ (or $d < d_0$ for some fixed integer $d_0$).

Similarly, you can replace his (right half-plane) condition that $E_1^{s,d} = 0$ and $\bar E_1^{s,d} = 0$ for $s < 0$ (or $s < s_0$ for some fixed integer $s_0$) by the (lower half-plane) condition that $E_1^{s,d} = 0$ and $\bar E_1^{s,d} = 0$ for $d > 0$ (or $d > d_0$ for some fixed integer $d_0$).

To see this, you can basically reindex the filtration so as to move the internal degree $d$ part of filtration $s$ to filtration $s+d$, keeping the internal degree. This shears the $E_1$-term, moving the vertical axis to the horizontal axis.

This adjustment is often enough. Do you need to refer to other half-planes than those bounded by a horizontal or a vertical line?

After a long life in preprint form, Boardman's paper was published in the conference proceedings celebrating his 60th birthday:

  \bib{MR1718076}{article}{
       author={Boardman, J. Michael},
       title={Conditionally convergent spectral sequences},
       conference={
          title={Homotopy invariant algebraic structures},
          address={Baltimore, MD},
          date={1998},
       },
       book={
          series={Contemp. Math.},
          volume={239},
          publisher={Amer. Math. Soc., Providence, RI},
       },
       date={1999},
       pages={49--84},
       review={\MR{1718076}},
       doi={10.1090/conm/239/03597},
    }

In the $(s,d)$-bigraded case, you can replace Boardman's (left half-plane) condition that $E_1^{s,d} = 0$ for $s > 0$ (or $s > s_0$ for some fixed integer $s_0$) by the (upper half-plane) condition that $E_1^{s,d} = 0$ for all $d < 0$ (or $d < d_0$ for some fixed integer $d_0$).

Similarly, you can replace his (right half-plane) condition that $E_1^{s,d} = 0$ and $\bar E_1^{s,d} = 0$ for $s < 0$ (or $s < s_0$ for some fixed integer $s_0$) by the (lower half-plane) condition that $E_1^{s,d} = 0$ and $\bar E_1^{s,d} = 0$ for $d > 0$ (or $d > d_0$ for some fixed integer $d_0$).

This adjustment is often enough. Do you need to refer to other half-planes than those bounded by a horizontal or a vertical line?