I guess you get the same extension problem if you just try to use the AHSS?

There are a couple more references, also related to the Picard group - the dual of the question mark complex is referenced in Goerss-Henn-Mahowald-Rezk's "Picard groups at chromatic level 2 for $p = 3$" paper - they at least tell you a bit about its $K$ theory and $KO$ theory. There are also some (hard!) calculations in Ichigi-Shimomura's "$E(2)_*$-invertible spectra smashing with the Smith-Toda spectrum $V(1)$ at the prime 3" (see section 3 in particular)

If you specifically want to see these in action, 'Bordism, Stable Homotopy, and Adams Spectral Sequences' by Kochman, does some serious $p=2$ calculations of the stable homotopy groups of spheres

Also, if you check out the work of Shimomura and colleagues, who use the chromatic spectral sequence in a fairly heavy way, I'm sure you'll find Massey products a plenty!

But how many people are really doing ASS calculations? If you look at, say, the calculation of the homotopy of tmf at $p=2,3$ then it makes heavy use of Massey products.

In particular, I hadn't realised that there was a spectral sequence $E_1^{n,k} = \bigoplus_{n=0}^2 \pi_k M_n(S) \Rightarrow \pi_k L_{E(2)} S$ for example

Good question, something I've been wondering about. Have you read Behren's "The homotopy groups of the $E(2)$-local sphere at $p > 3$, revisited"? There is a lot of good stuff in there! You might be interested in section 7 where he uses the calculations of $H^*(M_1^1)$ and $H^*(M_0^2)$ to calculate the homotopy of the $K(2)$ local sphere, the $E(2)$ local sphere and the $K(2)/E(2)$ local mod $p$ Moore spectrum.

Perhaps "Commutative Algebra: with a View Toward Algebraic Geometry" by Eisenbud is what you are after?