I am very glad this helped. When learning spectral sequences, working out a lot of examples is the key. On second thought, maybe it's not so different from anything else in mathematics.

I doubt there is anything significant you can say, other than what the spectral sequence will give you directly.

The spectral sequence linking the cohomology of the sheaves $K^i$ with hypercohomology is an $E_1$ spectral sequence; with the stated restrictions, it could have differentials at levels $E_1$ and $E_2$, I don't think you can extract a Gysin type exact sequence from it.

To handle the case that $E$ and $F$ are direct sums of line bundles, it is easier to restrict to a line; sums of line bundles are determined by their restrictions to a line. Also, the case that $E$ is a direct sum of copies of the same line bundle follows by the same trick from Horrock's theorem that a vector bundle with splitting type $(d, \dots, d)$ on all lines splits.

Yes, absolutely. Thinking about mathematics also involves writing discussing it with others. Examples are extremely important: start from the stupidest, most degenerated case you can think of, analyze it thoroughly, then move to slightly less stupid cases, and so on.

The fact that vector bundles on $\mathcal M_{1,1}$ split as sums of line bundles in characteristic larger than 3 can also be seen by the classical description of $\mathcal M_{1,1}$ as an open substack of the weighted projective stack $\mathbb P(4,6)$, coming from the Weierstrass form. Any locally free sheaf on $\mathcal M_{1,1}$ extends to a reflexive sheaf on $\mathbb P(4,6)$, which is locally free, because $\mathbb P(4,6)$ is regular of dimension 1. It is not hard to prove that any locally free sheaf on a weighted projective stack $\mathbb P(m,n)$ splits as a direct sum of line bundles.