You can find a proof in Hatcher's Algebraic Topology book - Proposition 4.56 on pp. 385

Not necessarily related to infinite families but in Hopkins ICM address he discusses what is known geometrically about the first 16 stable homotopy groups (arxiv.org/abs/math/0212397)

@Tyler: Thanks for that. So are you saying, in the example I have in mind, $d[\xi_1 \vert \xi_1^2\xi_2 ]=[\xi_1 \vert \xi_1^2 \vert \xi_2] + [\xi_1 \vert \xi_2 \vert \xi_1^2]+\cdots$?

@John - thanks that is pretty neat!

Ok, I've verified the relations in 2 now - thank you again. So it is really 'grunt work' - trying to find boundaries who have the right summand to produce a relation?

Thank you very much! I've finally managed to locate your thesis, so I'll sit down this morning and try and work through the calculations. (Hopefully I've fixed in the typo's in the first question now. I guess my problem in part 3 is - how does one do the coproduct $\psi(\xi_1^2 \overline{\xi}_2)$?)

@Sean - thank you, I have fixed the typo(s). I can do a resolution of $\mathbb{F}_2$ by hand for $\mathscr{A}(1)$, but I can't see how to do it for $\mathscr{A}(1)_*$ - I would be interested in seeing this!

Thanks Tyler! In its simplest form: $L_{E(n)}(E \wedge X) = L_{E(n)}(\mathbb{S}) \wedge E \wedge X = L_{E(n)} (E) \wedge X = E \wedge X$?