An event is scheduled here by me on the occasion of Karl Weierstrass' birthday. Please register if you are interested.

@AaronMazel-Gee The book in question is "handbook of model categories" which I just finished writing and is with referees at the moment. I guess it is kind of like "counter examples in topology" but model categories

Ironic that all these texts are called 'handbooks' when (e.g.) the Handbook of Homotopy Theory weighs 4.16 pounds and is nearly 1000 pages long

But I suppose it's text to have 'at hand' rather than 'in your hand'

None of them is a catalogue of hands, so once that band-aid is ripped off I think anything is possible

See e.g. chospab.es/biblioteca/DOCUMENTOS/Atlas_of_Hand_Bone_Age.pdf

Is there a formal-group-theoretic significance to the fact that the generator of the degree zero mod 2 cohomology of the Brown-Peterson spectrum BP (at p=2) exhibits the mod 2 homology of BP as the subalgebra of the dual Steenrod algebra generated by the squares of the usual generators?

There's probably deeper things that can be said, but one thing to notice here is that $H_{\bullet} BP$ (which is a subalgebra of the dual Steenrod algebra $H_{\bullet}H$) is isomorphic

$H_{\bullet} \otimes _{BP} BP_{\bullet} BP \otimes _{BP} H_{\bullet}$, which is the Hopf algebroid (in this case, even a Hopf algebra) corresponding to the automorphisms of the additive formal group, namely the formal group $Spf(H^{\bullet}(CP^{\infty}))$.

This is the more fundamental fact in the sense that this holds at all primes, not just at $p=2$.

From this perspective, I think it is a little bit more interesting to ask what is the significance of elements of $H_{\bullet} H$ that do not live in $H_{\bullet} HP$, as these do not a priori have interpretation as anything coming from formal groups, or in any case from the Quillen formal group $H^{\bullet} CP^{\infty}$.

At $p = 2$, an interesting thing happens, because instead of $H^{\bullet}(CP^{\infty})$ you can consider the formal group law on $H^{\bullet}(RP^{\infty})$. These two are in fact isomorphic if we disregard the grading (in terms of coefficients rings, through the Frobenius), and similarly $H_{\bullet}BP$ is isomorphic to $H_{*}H$, also through the Frobenius.

This way, you can identify $H_{*}H$ with representing automorphisms of $H^{\bullet}(RP^{\infty})$, but I would say that this is a consequence of the same fact for $H_{\bullet} BP$, which is the more fundamental object.

(A similar story plays out at odd primes, $H_{\bullet} H$ corepresents automorphisms [in an appropriate sense] of $Spf(H^{\bullet}(BC^{p}))$. It is, however, no longer true that $H^{\bullet}(BC^{p})$ and $H^{\bullet}(CP^{\infty})$ are isomorphic, even after disregarding the grading.)