Struggling with paracompact & countably many components $\implies$ second-countable.
We just need to show each component is second-countable. Choose a component $M_i$. $M_i$ is closed so it is paracompact. We can cover $M_i$ with precompact coordinate balls. (Note: A coordinate ball is an open set $U$ with a homeomorphism $\phi:U\rightarrow B$, where $B$ is an open ball in $\mathbb R^n$. A set is precompact if its closure is compact.) By paracompactness, we get a locally finite cover of precompact coordinate balls.
Now I am stuck. A previous exercise proved that, if $\mathcal X$ is a locally finite cover of precompact open sets, then any $X\in\mathcal X$ intersects only finitely many elements of $\mathcal X$. I am trying to use this fact to reduce the cover I obtained to a countable cover, thus proving the component is $\sigma$-compact (and therefore second countable).