Let us do the example where $\mathscr{M}$ is a proper Deligne-Mumford moduli stack of certain type of varieties such that the associated coarse space $M$ doesn't have any embedded point of codimension $1$. For exameple, this is the case where the varieties parameterized by $\mathscr{M}$ are stable curves of fixed genus or elliptic curves with some level structure as in Deligne-Rapoport's paper. Note that if $M$ is one-dimensional, the absence of embedded points in $M$ is equivalent to the Cohen-Macauleyness of $\mathscr{M}$. For simplicity, the objects parameterized by $\mathscr{M}$ will be called "curves".
A line bundle $\mathscr{L}$ over $\mathscr{M}$ in this case is an assignment of each family of curves $f : \mathscr{X} \to S$ to a line bundle $\mathscr{L}_f$ over $S$ such that whenever we have a fibered product
\begin{array}{c} \mathscr{X'} & \to & \mathscr{X} & \\ \downarrow& & \downarrow \\ S' & \xrightarrow{p} & S \end{array} the line bundles $\mathscr{L}_{f'}$ and $p^*\mathscr{L}_f$ coincide.
A rational section $s$ of $\mathscr{L}$ is thus naturally defined by picking a rational section $s_f$ of $\mathscr{L}_{f}$ for each $f$, compatible with $\mathscr{L}_{f'} = p^*\mathscr{L}_f$ for all $p$ as above. In particular, if $f : \mathscr{X} \to S$ is a family such that $S$ is smooth in codimension $1$ and the induced map $q : S \to M$ to the coarse space of $\mathscr{M}$ is proper, surjective, and generically finite, we have a rational section $s_f$ and thus a $\mathbf{Q}$-divisor class
$$\frac{1}{\deg(q)} q_*\mathrm{div}(s_f) \in \mathrm{Pic}(M)_\mathbf{Q}.$$
(Here we have to assume that $M$ doesn't have any embedded point of codimension $1$ to insure the existence of such an $f : \mathscr{X} \to S$). This divisor class is independent of the choice of $s$ and $f$, which is precisely the image of $\mathscr{L}$ under the pushforward map
$$\pi_* : \mathrm{Pic}(\mathscr{M})_\mathbf{Q} \to \mathrm{Pic}(M)_\mathbf{Q}$$ as Niels already mentioned in his comment.
When $\mathscr{M}$ is one-dimensional, the degree of $\mathscr{L} \in \mathrm{Pic}(\mathscr{M})_\mathbf{Q}$ that Deligne and Rapoport define is the degree of $\pi_* \mathscr{L}$, and your questions 2), 3) and 4) follow from the same statements for line bundles over schemes.