@Werner Thumann: That is correct, each prism manifold has exactly two Seifert fiberings. This is an interesting contrast to lens spaces, $S^3$, and $S^1\times S^2$, each of which has infinitely many different Seifert fiberings.

For question 1, I think there is a similar sort of argument to show injectivity in these cases, though it needs a bit more input, notably Adams operations in real K-theory. I have some handwritten notes on this from 20 years ago that would take some work to decipher after this long a time. My recollection is that I extracted these from Adams' J(X) - IV paper. My plan was, and still is, to include this in that unfinished book mentioned in the question, though as the years pass the chances of this ever happening become increasingly slim.

One can also reduce the case of nonempty boundary to the closed case by doubling: Take two copies of the surface with boundary and identify their boundaries to get a closed surface. If you know the closed surface is a torus, the original surface must then be an annulus. (Doubling doubles the Euler characteristic.)

The paper of LaBach mentioned in an earlier comment (now deleted?) uses a nonstandard topology on $Diff(D^n)$. The restriction map $Diff(D^n)\to Diff(int(D^n))$ is injective so can be viewed as an inclusion, and LaBach uses the subspace topology on $Diff(D^n)$ induced from the compact-open topology on $Diff(int(D^n))$. In this topology one can do a sort of "reverse Alexander trick" and push all the complications of a diffeomorphism of $D^n$ out to $\partial D^n$ and make them disappear.

$Diff(D^n)$ is homotopy equivalent to $O(n)\times Diff(D^n\ rel\ D^{n-1})$ where $D^{n-1}$ is a disk in $\partial D^n$. The factor $Diff(D^n\ rel\ D^{n-1})$ can be identified with the pseudoisotopy space $Diff(D^{n-1}\times I\ rel\ D^{n-1} \times 0)$, so it has a complicated homotopy type when $n$ is large enough.