Five term exact sequence of a first quadrant homological spectral sequence

At $(1,0)$ the outgoing target $(1-r,r-1)$ and incoming source $(1+r,1-r)$ are zero. Thus $E_{1,0}^2$ is canonically $E_{1,0}^{\mathrm{\infty }}=H_1\mathrm{/}{F}_0{H}_1$, and the edge $H_1\to E_{1,0}^2$ is the epic quotient map.

At $(0,1)$ all outgoing targets $(-r,r)$ vanish. Its incoming source is $(r,2-r)$, which lies in the quadrant only for $r=2$. Therefore $E_{0,1}^{\mathrm{\infty }}=\mathrm{coker}(d^2:E_{2,0}^2\to E_{0,1}^2)$, identified by abutment with $F_0{H}_1\mathrm{/}{F}_{-1}{H}_1=F_0{H}_1$. The edge $E_{0,1}^2\to H_1$ is the cokernel projection followed by the filtration inclusion.

At $(2,0)$ all incoming sources $(2+r,1-r)$ vanish. Its outgoing target $(2-r,r-1)$ lies in the quadrant only at $r=2$. Hence $E_{2,0}^{\mathrm{\infty }}=\ker d^2$, identified with $F_2{H}_2\mathrm{/}{F}_1{H}_2=H_2\mathrm{/}{F}_1{H}_2$. The edge $H_2\to E_{2,0}^2$ is the quotient onto that kernel followed by its monic inclusion.

Step 1.3 proves that the image at $E_{2,0}^2$ is $\ker d^2$. Step 1.2 proves that the next kernel at $E_{0,1}^2$ is $\mathrm{im}{d}^2$, and its image in $H_1$ is $F_0{H}_1$. This is the kernel of the quotient in step 1.1, which is onto, proving exactness also at $E_{1,0}^2$ before the terminal zero. These are precisely all claimed positions. There is no claim about a kernel at the initial $H_2$ without a preceding map. Zero terms and zero $d^2$ give the same kernel and cokernel factorizations; the degree-one normalized endpoints are used explicitly. All maps are canonical from the specified data, without splittings or AC.