good point! it is not completely clear to me yet... :D just kidding: as I have edited, the reasons for which there are different actions of the modular groups on characteristics (one or two orbits) -depending on the level - is still unclear to me.

Yes I meant the finite field of order 2, thank you! What is the precise book reference that you suggest?

no, I am sorry, I should have explained. By even and odd I mean the Arf invariant of a quadratic form on a $\mathbb{Z}_2$-vector space. I will correct the question.

Very nice proof. As far as I understand the same proof would not hold for even $d$, right?

There's a cute subgroup of $Sp(4, \mathbb{Z}_2)$ of index 6, which is the stablizer of an odd theta-characteristic (seen as a quadratic form on the 2-torsion point). I wonder if it is this one.