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.
Use Kuratowski embedding to embed your metric space isometrically into a Banach space.