Let us do the example where $\mathscr{M}$ is a proper Deligne-Mumford moduli stack of varieties. Note that if $\mathscr{M}$ is one-dimensional, then the Cohen-Macauleyness of $\mathscr{M}$ is equivalent to the absence of embedded points in $M$; this assumption together with the properness is to insure that the degree map $\mathrm{Pic}(M) \to \mathbf{Z}$ is well defined.
A line bundle $\mathscr{L}$ over $\mathscr{M}$ in this case is an assignment of each family $f : \mathscr{X} \to S$ of varieties parameterized by $\mathscr{M}$ 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. The first Chern class of $\mathscr{L}$ is defined by picking a rational section $s_f$ of $\mathscr{L}_{f}$ for each $f$ such that $\mathrm{div}(s_{f'}) = p^*\mathrm{div}(s_{f})$ in $\mathrm{Pic}(S')$ for all $p$ as above.
One can also define a rational section $s$ of $\mathscr{L}$ by picking a rational section $s_f$ of $\mathscr{L}_{f}$ for each $f$ such that $S \to \mathscr{M}$ is étale, compatible with $\mathscr{L}_{f'} = p^*\mathscr{L}_f$ for all $p$ as above which are necessarily étale. The $\mathbf{Q}$-divisor class
$$\frac{1}{\deg(q)} q_*\mathrm{div}(s_f) \in \mathrm{Pic}(M)_\mathbf{Q}$$
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.