Yes, I think now both questions have a positive answer. Denote the cokernel of $N\hookrightarrow M$ by $T$. This is a torsion sheaf with first Chern class $c_1(T)=c_1(M)-c_1(N)=0$. So it is supported in codimension two, but that means $M$ and $N$ only differ in codimension two. As they are locally free, they must already be isomorphic. For the second question: the injection $N\hookrightarrow M$ induces an injection of determinant line bundles. This gives the desired inequality.

I think some of what you are looking for can be found in the proof of Theorem 1 in Section 4 of the paper " Moduli of Vector Bundles on a Compact Riemann Surface" by Narasimhan and Ramanan.

For g=2, r=2 and fixed determinant $\mathcal{O}_X$ the moduli space is isomorphic to $\mathbb{P}^3$ and the semistable locus is the singular Kummer surface associated to $X$, hence has codimension 1. This all explained in the papers by Narasimhan et. al.

No in this case we have $R^1(pr_2)_{*}F=0$ which implies $H^1(F_u)=0$ for the sheaves on the fiber over a point $u\in U$. As I said, I don't really know what is meant here, only some ideas. But look at arXiv:1602.05267 on the beginning of p.17. There the authors do it exactly as I think. As one of the authors is Neumann, maybe you can just ask him, what is meant in the book?

@Jason Starr: Thank you. Looking through some papers it seems that the Gysin sequence is functorial in the pair $(X,D)$. But I could not find a proof of this fact, it is only mentioned. Do you know a reference? I could not find it in Grothendieck's articles in Dix Exposes.