Rays in non-compact spaces According to the book Infinite homotopy theory by Baues-Quintero, if $X$ is a locally compact, locally connected,  connected metrizable space, its Freudenthal ends can be identified (Proposition 9.20) by certain equivalence classes of rays. 
In particular, if $X$ is non-compact, there is a proper map $r:[0,\infty) \rightarrow X$.  1) Can $r$ be chosen so that it is also an embedding? 2) If yes, does $X$ have a metric for which $r$ can be chosen as an isometric embedding?
 A: The answer to both questions is yes.
For the first question, recall that if $W$ and $X$ are non-compact
locally compact
Hausdorff spaces, a mapping $f: W \to X$ is continuous and proper iff
the mapping between their respective one point compactifications 
$\overline{f}: W \cup \{\infty\} \to X \cup \{\infty\}$ with 
$\overline{f}(\infty) = \infty$ and $\overline{f}|_W = f$ is continuous.
In particular, if $W = [0,\infty]$ there is a path in $X\cup\{\infty\}$
from $f(0)$ to $\infty$. The image of this path is a Peano space and
therefore arc-connected, so there is an arc from $f(0)$ to $\infty$.
After appropriate re-parametrisation and restriction this gives us a
proper continuous injection $r: [0, \infty) \to X$. Since a proper
continuous mapping between locally compact Hausdorff spaces is closed,
this is a homeomorphic embedding.
The fact that the embedding is closed also provides the answer to the second question, as any compatible metric on a closed subspace of a
metrizable  space can be extended to the whole space. If desired,
the extension can be complete, as described in
Bacon, P. (1968). Extending a Complete Metric. The American Mathematical Monthly, 75(6), 642-643.
