The space $\mathcal{S}$ of rapidly decaying functions is endowed with its usual Fréchet topology. Its topological dual $\mathcal{S}'$ is endowed with the strong topology.
I assume I have a norm $\lVert \cdot \rVert$ on $\mathcal{S}$ that is continuous over $\mathcal{S}$ and denote by $\mathcal{X}$ the Banach space obtained as the completion of $(\mathcal{S},\lVert \cdot \rVert)$.
Is it correct to say that we have the embeddings $$\mathcal{S}\subseteq \mathcal{X} \subseteq \mathcal{S}'?$$ What if $\lVert \cdot \rVert$ is only a semi-norm?
No, this isn't correct. To make things easier consider the "discrete" case of the rapidly decreasing sequences $s$ so that $s'$ is the space of slowly increasing sequences. As a norm on $s$ take $\|x\|= \sum_{n=1}^\infty |x_n|/e^n$. The completion of $(s,\|\cdot\|)$ is then the weighted $\ell^1$-space of all sequences with $\sum_{n=1}^\infty |x_n|/e^n$ finite which is not contained in $s'$.
The point is, that the inclusion $(s,\|\cdot\|) \hookrightarrow s'$ is not continuous. If $\|\cdot\|$ is only a seminorm, $\cal X$ should be the Hausdorff completion of $(s,\|\cdot\|)$ and the canonocal map $s\to \cal X$ will no longer be injective.