Converting graphs to sets of paths
This should be NP-hard, here's a way you could try to do a reduction:

Take any graph $G$ embedded in the $\mathbb{R}^2$ plane. Let $c_{min} = \min_{e \in E}\{c(e)\}$  be the minimum edge weight in the graph ($c$ is the weight function of the edges of $G$, i.e. the euclidean distance). To build $G'$, split every vertex $v = (x,y)$ into $v_1 = v$ and $v_2 = (x+\epsilon,y)$ where $\epsilon = \frac{c_{min}}{2^n}$, connect $v_1$ and $v_2$ with an edge weighted by their euclidean distance. Obviously, no vertex in $G'$ is incident with more than two edges, and there are no vertices that are incident with no edges.

What you now would have to prove is something like "there's a TSP route of weight $\le w$ in $G$ iff there's a TSP route of weight $\le w + n \cdot \epsilon$ in $G'$. I don't know how to prove that, but some geometry and use of the triangle inequality might help. I expect the choice of $\epsilon$ as vanishingly small will be enough to give the reduction.

EDIT: The reduction will remain valid after your edit.