Show that unit circle is not homeomorphic to the real line

You can certainly show these using compactness. The following proofs, however, I find simpler:

The removal of any one point from $\Bbb R$ results in a disconnected space, but if you remove one point from $S^1$, you still have a connected space.

The removal of any two points from $S^1$ results in a disconnected space, but if you remove two points from $\Bbb R^2$, you still have a connected space.

To prove that $\mathbb R^2$ is not compact: Assume that it is. The image of a compact space under a continuous map is compact. The mapping $f:\mathbb R^2 \to \mathbb R, (x,y)\mapsto x$ is continuous and has image $\mathbb R$. Hence $\mathbb R$ is compact. But you yourself showed that $\mathbb R$ is not compact. Contradiction.

To show that $S^1$ is not homeomorphic to $\mathbb R^2$: Observe that $S^1$ is compact but $\mathbb R^2$ isn't. Done.