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Such decompositions are not unique, as (for example) $Sq^6 = Sq^2Sq^4 + Sq^1Sq^2Sq^1Sq^2$ as well. Sage can do these types of change of basis for you. But you might also find what you want in mathweb.scranton.edu/monks/pubs/bases.pdf (Look at Wall's basis and Arnon's A basis)
There should be more extensions in Ext^1 - it's equivalence classes of extensions of M by N. e.g., 0 --> Z/2 ---> Z/4 ---> Z/2 --> 0 corresponds to the non-zero element of Ext^1(Z/2,Z/2)
When using Lemma 2.1, aren't you making the assumption that (i) Ext_{dw}^1(X,\Sigma^{-n-1}Y) = \Ext_{dw}^n(X,Y) (for whatever the latter means, since it is not defined in the cited paper) (ii) That Ext_{dw}^n(M,N) = \Ext_R^n(M,N).
@user51223 See the bottom of page 4 of arxiv.org/pdf/1407.8418.pdf - there are some inconsistencies with the latest calculations and the Kochman/Mahowald calculations. Thus, I'm not sure if the calculation in the 64 stem is correct.
The usual reference for these types of things is Boardman's 'Conditionally convergent spectral sequences' - citeseerx.ist.psu.edu/viewdoc/…. The remark after Theorem 7.1 suggests the answer to your question is that such a spectral sequence strongly converges.