_The original question:_
> Is there a name for a exactly one vertex with exactly one edge in graph terminology?
It's a node of degree/valence $1$, sometimes called a _leaf_ or an _end_.
This all appears in the wiki on degree (graph theory)).
> Is there a name for a node and an in- or outgoing edge from that node as a unity[...]?
I guess you mean "as a unit" as in "one term for the two things together." I don't see why anyone would need something more complicated than "an edge $E$ incident on $V$" or "endpoint $V$ of edge $E$."
To me, endpoints are already fully integrated into their edge. What is an edge without its endpoints? Can you even talk about them "not as a unit?"