Yes, there are such things.

Consider the monoid $M$ of endomaps $\{0,1\} \to \{0,1\}$. It has four elements: the constant maps 0 and 1, the identity map $i$ and the "swap" map $s$. Let $\mathcal{G}$ be the category of right $M$-actions. These are sets equipped with a map $m : A \times M \to A$ such that $m(x, i) = x$ and $m(x, f \circ g) = m(m(x, f), g)$. A morphism is a map which commutes with the actions. I will write $x \cdot f$ instead of $m(x, f)$. (All this is very common, except that we have a monoid instead of a group.)

Let us think of the elements of an $M$-set $A$ as half-edges. Each half-edge $e \in A$ is connected to its opposite half-edge $e \cdot s$. Call a half-edge $e$ a vertex when $e \cdot 0 = e$ (exercise: $e \cdot 0 = e \iff e \cdot 1 = e$). Each half-edge $e$ has an origin which is $e \cdot 0$, while $e \cdot 1$ is the origin of its opposite half-edge $e \cdot s$. It is possible to have a half-edge which is its own opposite, $e \cdot s = e$ (these are your dangling edges). The graphs are reflexive because every vertex $v$ has a distinguished half-edge attached to it, namely $v$ itself.

Explicitly, we have the following equivalent presentation of $\mathcal{G}$:

1. objects are sets $(V, H)$ of vertices and half edges such that:

   • each half-edge $e \in H$ has an origin $o(e) \in V$
   • each half-edge $e$ has an opposite half-edge $s(e) \in E$
   • each vertex $v \in V$ has a distinguished half-edge $\ell(v) \in H$ such that $o(\ell(v)) = v$ and $s(\ell(v)) = \ell(v)$

2. A morphism $(V,H) \to (V',H')$ is a pair of maps $f : V \to V'$ and $g : H \to H'$ such that $g(\ell(v)) = \ell(f(v))$, $f(o(e)) = o(g(e))$, and $s(g(e)) = g(s(e))$. Morphisms are composed component-wise.

This an more can be read about in:

William F. Lawvere, Qualitative distinctions between some toposes of generalized graph, Contemporary mathematics, Vol. 92, 1989, pp. 261-299.

Yes, there are such things.

Consider the monoid $M$ of endomaps $\{0,1\} \to \{0,1\}$. It has four elements: the constant maps 0 and 1, the identity map $i$ and the "swap" map $s$. Let $\mathcal{G}$ be the category of right $M$-actions. These are sets equipped with a map $m : A \times M \to A$ such that $m(x, i) = x$ and $m(x, f \circ g) = m(m(x, f), g)$. A morphism is a map which commutes with the actions. I will write $x \cdot f$ instead of $m(x, f)$. (All this is very common, except that we have a monoid instead of a group.)

Let us think of the elements of an $M$-set $A$ as half-edges. Each half-edge $e \in A$ has an associated opposite half-edge $e \cdot s$. Call a half-edge $e$ a vertex when $e \cdot 0 = e$ (exercise: $e \cdot 0 = e \iff e \cdot 1 = e$). Each half-edge $e$ has an origin which is $e \cdot 0$, while $e \cdot 1$ is the origin of its opposite half-edge $e \cdot s$. It is possible to have a half-edge which is its own opposite, $e \cdot s = e$ (these are your dangling edges). The graphs are reflexive because every vertex $v$ has a distinguished half-edge attached to it, namely $v$ itself.

Explicitly, we have the following equivalent presentation of $\mathcal{G}$:

1. objects are sets $(V, H)$ of vertices and half edges such that:

   • each half-edge $e \in H$ has an origin $o(e) \in V$
   • each half-edge $e$ has an opposite half-edge $s(e) \in E$
   • each vertex $v \in V$ has a distinguished half-edge $\ell(v) \in H$ such that $o(\ell(v)) = v$ and $s(\ell(v)) = \ell(v)$

2. A morphism $(V,H) \to (V',H')$ is a pair of maps $f : V \to V'$ and $g : H \to H'$ such that $g(\ell(v)) = \ell(f(v))$, $f(o(e)) = o(g(e))$, and $s(g(e)) = g(s(e))$. Morphisms are composed component-wise.

This an more can be read about in:

William F. Lawvere, Qualitative distinctions between some toposes of generalized graph, Contemporary mathematics, Vol. 92, 1989, pp. 261-299.