Question: Is every metric space a subspace of some connected metric space? If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected. I'm not sure whether this will work or not, but intuitively I think the answer is yes.

Answer: 
If we just want $(X,d)$ to be any subspace of a larger connected space, then sure: we don't even need metric then, just Tychonoff: every Tychonoff space embeds into a (connected) space $[0,1]^I$ for some set $I$.

If we want $X$ to be dense in the larger connected space $Y$, then $Y$ is called a connectification. In that case there are examples of metric spaces without a metric connectification (see the encyclopedia of general topology, chapter on connectifications), for references and some positive results as well.
Here is another way to embed any metric space $X$ in a connected metric space. For each pair of distinct points $x,y\in X$, adjoin to $X$ a path from $x$ to $y$, and extend the metric to points on the path by making the path have length $d(x,y)$ (so distances are defined by taking the least distance which can be achieved along the paths or through $X$; by making the paths have length $d(x,y)$, distances between points of $X$ will be unchanged). The result will be a metric space in which $X$ embeds isometrically which is connected (in fact, path-connected).