Higher level analogs of Nicolas-Serre theory
When N=3, and V(plus) and D are as in my question, I now believe that I have the right "code" for monomials in V(plus) and that it should be possible to use this code to yield:
(*)-- "If HE# is the algebra spanned by the operators T_n with n=1 mod 6, and m is the ideal Ann(D) in HE#, then the m-completion of HE# acting on V(plus) is a 2-variable power series ring in T_7 and T_13."
__THE NICOLAS-SERRE CODE
__ I'll begin by describing the Nicolas-Serre code for monomials in the subspace V of Z/2[[x]] spanned by the F^k, k odd, where F=x+x^9+x^25+x^49+... If c is in N, write c as a sum of distinct powers of 2, and let c2 be the sum of the squares of these powers. If (a,b) is in NxN, the "corresponding monomial in V" is F^k where k=1+2(a2)+4(b2). We say that the code of D^k is (a,b) and refer to D^k as [a,b]. Now there is a listing in order of the elements of NxN: (0,0) /(1,0),(0,1) /(2,0),(1,1),(0,2) /(3,0),(2,1),(1,2),(0,3) /... where I've written slashes where a+b jumps. This gives a corresponding listing in order of the monomials in V: D /D^3,D^5 /D^9,D^7,D^17 /D^11,D^13,D^19,D^21 /... In Propositions 4.3 and 4.4 of their paper on the order of nilpotence, Nicolas and Serre say:
__If a>0, T_3 takes [a,b] to [a-1,b]+ a sum of earlier terms.
__T_3 takes [0,b] to a sum of [i,j], with i+j < b.
__If b>0, T_5 takes [a,b] to [a,b-1]+ a sum of earlier terms.
__T_5 takes [a,0] to a sum of [i,j] with i+j < a.
---They haven't written out their proofs in the article -- they say it's long and technical and depends on certain recurrences that they explicitly give. Once the propositions are established, a fairly simple argument shows that the completion of the Hecke algebra HE acting on V, with respect to the maximal ideal Ann(F), is a power series ring in T_3 and T_5.
--THE LEVEL 3 CODE
__Now I attach to the element (a,b) of NxN the monomial D^k in V(plus) where k is 1+6(a2)+12(b2). The "code" of D^k is (a,b) and I refer to D^k as [a,b]. As above we get a listing in order of the monomials in V(plus): D /D^7,D^13 /D^25,D^19,D^49 /D^31,D^37,D^55,D^61 /... Experimentally I find that when a+b < 7 the results of Propositions 4.3 and 4.4 quoted above hold if we replace T_3 and T_5 by T_7 and T_13. And there are recursions analogous to those used by Nicolas and Serre, coming from modular equations. Deducing the analogs to the Nicolas-Serre propositions from these recursions is no doubt "long and technical" in spades, but it should be doable, and (*) from the beginning of this answer ought then to follow.
EDIT:
I now believe I can handle my N=3 conjectures, and indeed show that a certain completed Hecke algebra identifies with Z/2[[X,Y]] with an element of square 0 adjoined. Surprisingly I only need the recursions for T_7, together with information about the action of T_13 on a certain space of higher level weight one modular forms annihilated by T_7.
Here's an outline starting with another proof of the Nicolas-Serre level one theorems. Let F in Z/2[[x]] be the reduction of delta, and V be spanned by the V^k, k odd. There is a Z/2[[X,Y]]-action on V with X and Y acting by T_3 and T_5. One wants to explicitly describe V as Z/2[[X,Y]]-module (and determine the smallest k such that (X,Y)^k annihilates any given F^n).
Nicolas and Serre use their technical Propositions 4.3 and 4.4 to do this. 4.3, based on a recursion for the T_3(F^n), shows that T_3 is onto and that the kernel is "not too big" while 4.4 does the same for T_5. Mathilde Gerbelli-Gauthier has given a clear proof of 4.3, but no good proof of 4.4 is known--the Nicolas-Serre proof is long, technical and unpublished.
I circumvent 4.4 as follows. If q is a power of 2, consider positive primitive binary forms whose discriminant is -64(q^2); their SL(2,Z)-classes form a cyclic group of order 2q under Gaussian composition. Modifying the theta series attached to such a form and reducing mod 2 yields an element of Z/2[[x]]. The modified theta series are weight 1 modular forms of level a power of 2 and it turns out that each of them is congruent mod 2 to an element of V. The reductions span a space of dimension q+1, annihilated by X. Using Gaussian composition one shows that this space is stabilized by Y and is a cyclic Z/2[[Y]]-module.
Combining this with the fact that the kernel of X is "not too large" one finds that this kernel is spanned by the reductions of modified theta series and is a direct limit of cyclic Z/2[[Y]]-modules. The rest of the argument is formal and not hard--one determines the structure of V as Z/2[[X,Y]-module, shows that each T_p acts on V by multiplication by an element of (X,Y), and concludes that the completed Hecke algebra identifies with Z/2[[X,Y]].
N=3 works out a little differently. In place of F we use D=F(x)+F(x^9). V is replaced by V(minus), spanned by the D^n with n=5 mod 6. V(minus) turns out to be a Z/2[[X,Y]]-module with X and Y acting by T_7 and T_13; the proof uses level 3 modular forms. The linear map, f(t)-->(D^2)f(D^3) identifies the space of odd power series in t with V(minus). Let A(n) correspond to D^n. Using the level 7 modular equation for F one finds:
(*)___ A(n+16)=(t^16)A(n)+(t^4)A(n+4)+(t^2)A(n+2).
This is like the recursion used to prove 4.3 (but the (t^2)A(n+2) term is absent there). The initial conditions are favorable, and Gerbelli-Gauthier's argument is easily modified to show that X is onto with "not too large" kernel. One then uses modified theta series (attached to the same binary forms as above) to show that the kernel is spanned by the reductions of such series and is a direct limit of cyclic Z/2[[Y]]-modules. We then get a complete description of V(minus) as Z/2[[X,Y]]-module and a proof that each T_p with p=1 mod 6 acts on V by multiplication by an element of (X,Y). An identical argument works for V(plus).
The T_p with p=5 mod 6 don't act on V(minus) though. So to study the full Hecke algebra one looks at the sum, W, of V(minus) and V(plus). Now (T_5)^2 does act on V(minus) and is multiplication by g^2 for a certain g in (X,Y). Now if we make W into a Z/2[[X,Y,U]]-module with X,Y and U acting by T_7, T_13 and T_5 + g(X,Y), we get a complete description of W as Z/2[[X,Y,U]]-module, and conclude that the completed Hecke algebra identifies with the quotient Z/2[[X,Y,U]]/U^2.
EDIT(9/1/15)
I write out in detail the argument sketched above for N=3 in an article: "A Hecke algebra attached to mod 2 modular forms of level 3". (arXiv.org/1508.07523)
FINAL EDIT--This is now all superseded by the results in the six arXiv articles referenced in my other (now accepted) answer. See the articles, or the brief summary of them given in the other answer.