Consider a digraph with two vertices $v_1$ and $v_2$, loops $\ell_{1}=e_{11}$, and $\ell_{2}=e_{22}$, and directed edges $e_{12}$ and $e_{21}$.
I am interested in finding the number of paths of length $k$ from vertex $v_1$ to $v_2$, so a series of edges:
$$x_1x_2\cdots x_k\cong e_{12}.$$
Suppose such a path traverses $\ell_i$ $c_i$ times, $e_{12}$ $d_1$ times, and $e_{21}$ $d_2$ times. Then $d_1=d_2+1$ and so:
$$d_2=\frac{k-c_1-c_2-1}{2}.$$
What I am looking for is a formula, in terms of $k$, $c_1$, and $c_2$, for the number of paths of length $k$ from $v_1$ to $v_2$ with $c_1$ loops $\ell_{1}$ and $c_2$ loops $\ell_2$.
I can solve this problem when the loop $\ell_2$ is deleted. The answer turns out to be:
$$\binom{\frac{k+c_1-1}{2}}{c_1}.$$
I am not sure if there is a better way of attacking the problem using multinomials, perhaps in terms of total loops, $c=c_1+c_2$ (i.e. a formula for the total number of paths from $v_1$ to $v_2$ with $c$ loops)?
I could push on with this but I have a feeling that generating functions might be the machinary to use... and unfortunately I am not au fait with them. Consider the following set up.
The path consists of $0\rightarrow k-1$ loops $\ell_1$, followed by a first $e_{12}$. Then there are $0\rightarrow k-1$ edges and loops $x_1x_2\cdots x_i$. Followed by a final $e_{12}$. Followed by $0\rightarrow k-1$ loops:
$$\ell_1\cdots \ell_{1}e_{12}\,x_1\cdots x_i\,e_{12}\,\ell_2\cdots \ell_2.$$
This is where I feel like a generating function approach may work.
Given $k, c_1, c_2$ (and thus also $d_1 = \frac{k-c_1-c_2+1}{2}$ as you already noted) the set of paths from $v_1$ to $v_2$ of length $k$ is in bijection to the set $$\{(a_1, b_1, a_2, b_2, \ldots, a_{d_1}, b_{d_1}) \in \mathbb{N}^{2d_1} | \sum_i a_i = c_1, \sum_i b_i = c_2\}$$ where $a_i$ counts the number of times you use $\ell_1$ directly before traversing $e_{12}$ the $i$-th time and $b_i$ counts the number of times you use $\ell_2$ directly after.
This set can be written as $$\{(a_1, \ldots, a_{d_1}) | \sum_i a_i = c_1\} \times \{(b_1, \ldots, b_{d_1}) | \sum_i b_i = c_2\}$$
whose cardinality is $$\binom{c_1+d_1-1}{c_1} \cdot \binom{c_2 + d_1 - 1}{c_2}$$ by a standard argument known as "stars and bars".