Definition A coherent sheaf $\mathcal{F}$ on an algebraic variety $X$ is nef if the following condition holds: For every irreducible curve $C\subset X$, the line bundle $O(1)$ is nef on $\mathbb{P}(\mathcal{F}|_C)$.
I haven't seen this definition of a nef coherent sheaf either, but I think most of the properties you mention just follow formally from the properties of ordinary nefness. Here is a proof for a) and b):
a) A quotient of a nef sheaf is nef. Let $C$ be as in the definition. If $F\to E\to 0$ is a surjection, this restricts to a surjection $F|_C\to E|_C\to 0$ and hence gives an embedding $\mathbb{P}(\mathcal{F}|_C)\hookrightarrow \mathbb{P}(\mathcal{F}|_C)$ such that $O(1)$ on \mathbb{P}(\mathcal{F}|_C)$\mathbb{P}(\mathcal{F}|_C)$ restricts to $O(1)$ on \mathbb{P}(\mathcal{E}|_C)$\mathbb{P}(\mathcal{E}|_C)$. Since the restriction of a nef line bundle is nef, $E$ is also nef.
b) PullbackA pullback of nef is nef. Similarily, if $f:X\to Y$ is finite, then $f$ restricts to a finite map $f:C'=f^{-1}C\to C$ and hence there is a finite map $F:\mathbb{P}(f^*\mathcal{F}|_C)\to \mathbb{P}(\mathcal{F}|_C)$ such that $O(1)=F^*O(1)$. Now the claim just follows from the corresponding statement for line bundles.