Struggling with recovering the associated graded vector space in A User's Guide to Spectral Sequences I've made it through a grand total of 1 page of John McCleary's A User's Guide to Spectral Sequences before getting stuck.
Let $H^*$ be a graded $k$-vector space, and that it is filtered
$$H^*= F^0H^*\supset \dots\supset F^nH^*\supset F^{n+1}H^* \supset \dots \supset \{0\}$$
then the associated graded vector space $E^p_0(H^*)$ is defined by $E^p_0(H^*) = F^pH^*/F^{p+1}H^*$. We can give this a bigrading by defining $F^pH^r = F^pH^*\cap H^r$ and then
$$E^{p,q}_0 = F^pH^{p+q}/F^{p+1}H^{p+q}$$
On page 4, the following is written about the definition of $E^{p,q}_0$

The associated graded vector space $E^p_0(H^*)$ can be recovered by taking the direct sum of the spaces $E^{p,q}_0$ over the index $q$

So taking the direct sum over $q$ we get
\begin{align*}
\bigoplus_{q=0}^\infty E^{p,q}_0 &= \frac{F^pH^p}{F^{p+1}H^p}\oplus \frac{F^pH^{p+1}}{F^{p+1}H^{p+1}}\oplus\frac{F^pH^{p+2}}{F^{p+1}H^{p+2}}\oplus\dots\\
&=\frac{\oplus_{q=0}^\infty F^pH^*\cap H^{p+q}}{\oplus_{q=0}^\infty F^{p+1}H^*\cap H^{p+q}}
\end{align*}
but I have no idea how to identify
$$\frac{\oplus_{q=0}^\infty F^pH^*\cap H^{p+q}}{\oplus_{q=0}^\infty F^{p+1}H^*\cap H^{p+q}}$$
with
$$\frac{F^pH^*}{F^{p+1}H^*}$$
or if it's even possible to do it. I have a feeling that I've made an error somewhere. I'm not entirely certain if I'm justified in assuming that $q$ is non-negative for instance.
As a bonus problem, I'm also not sure about the sentence immediately after the one I quoted above

To recover $H^r$ directly, as a vector space, take the direct sum $\bigoplus_{p+q=r}E^{p, q}_0$

so taking the example of $H^2$ we have
$$\bigoplus_{p+q=2}E_0^{p, q} = E_0^{0, 2}\oplus E_0^{1,1}\oplus E_0^{2, 0}=\frac{F^0H^2}{F^1H^2}\oplus\frac{F^1H^2}{F^2H^2}\oplus\frac{F^2H^2}{F^3H^2} \simeq \frac{H^2}{F^3H^2}$$
which clearly isn't isomorphic with $H^2$.
What am I missing here?
 A: Remember $H^*$ is a graded vector space, so there are no extension issues. Also, there are two gradings here, which makes it a bit harder to talk about. For the sake of communicating, let's call these the $H$-grad and $F$-grade.

Let's say $H^*=H^0\oplus H^1\oplus H^2$. So, for example, we can call $H^2$ the $H$-grade-$2$ term of $H^*$.
Now suppose we have another filtration
$$H^*=F^0H^*\supset F^1H^*\supset F^2H^*\supset F^3H^*\supset F^4H^*=\{0\}.$$
This gives the $F$-grading
$$H^*=\frac{F^0H^*}{F^1H^*}\oplus \frac{F^1H^*}{F^2H^*}\oplus \frac{F^2H^*}{F^3H^*}\oplus \frac{F^3H^*}{F^4H^*}$$
So $F^1H^*/F^2H^*$ is the $F$-grade-$1$ term of $H^*$.
Now the point is that $F$ also induces a grading on each of the $H^i$, so
$$H^0=\frac{F^0H^0}{F^1H^0}\oplus \frac{F^1H^0}{F^2H^0}\oplus \frac{F^2H^0}{F^3H^0}\oplus \frac{F^3H^0}{F^4H^0}$$
as well as
$$H^1=\frac{F^0H^1}{F^1H^1}\oplus \frac{F^1H^1}{F^2H^1}\oplus \frac{F^2H^1}{F^3H^1}\oplus \frac{F^3H^1}{F^4H^1}$$
and
$$H^2=\frac{F^0H^2}{F^1H^2}\oplus \frac{F^1H^2}{F^2H^2}\oplus \frac{F^2H^2}{F^3H^2}\oplus \frac{F^3H^2}{F^4H^2}.$$
Since $F$ is a filtration on the vector space $H^*$, note that it is more correct to write $F^iH^*\cap H^j$ rather than $F^iH^j$. All together, $H^*$ can be split into a $4\times 3$ grid, which motivates the way we write spectral sequences in pages.
