Why is multiplication on the space of smooth functions with compact support continuous?
You can spare yourself the functional analytic abstract nonsense by using an explicit set of seminorms on $\mathcal{D}(\mathbb{R}^d)=C_{c}^{\infty}(\mathbb{R}^d)$ which, unfortunately, are not well-known but can be found in the excellent book "Topological Vector Spaces and Distributions" by Horváth on p.171.
Let $\mathbb{N}=\{0,1,\ldots\}$, and denote the set of multiindices by $\mathbb{N}^d$. A locally finite family $\theta=(\theta_{\alpha})_{\alpha\in\mathbb{N}^d}$ of continuous functions $\mathbb{R}^d\rightarrow \mathbb{R}$ is one such that for all $x\in\mathbb{R}^d$ there is a neighborhood $V$ such that $V\cap {\rm Supp}\ \theta_{\alpha}=\varnothing$ for all but finitely many $\alpha$'s. Let $$ \|f\|_{\theta}=\sup_{\alpha\in\mathbb{N}^d}\sup_{x\in\mathbb{R}^d} |\theta_{\alpha}(x)D^{\alpha}f(x)|\ , $$ then the seminorms $\|\cdot\|_{\theta}$ where $\theta$ runs over all such locally finite families define the topology of $\mathcal{D}(\mathbb{R}^d)$.
Continuity of the pointwise product follows once you show that for every $\theta$, there exists $\theta'$ and $\theta''$ such that $$ \|fg\|_{\theta}\le \|f\|_{\theta'}\|g\|_{\theta''} $$ for all test functions $f$ and $g$, which one can do by hand.
For instance, you can use the Leibniz or product rule $$ D^{\alpha}(fg)=\sum_{\beta+\gamma=\alpha}\frac{\alpha!}{\beta!\gamma!} D^{\beta}f D^{\gamma}g\ , $$ and the brutal $l^1$-$l^{\infty}$ estimate $$ |D^{\alpha}(fg)|\le \prod_{i=1}^{d}(\alpha_i+1) \times\max_{\beta+\gamma=\alpha} \frac{\alpha!}{\beta!\gamma!} |D^{\beta}f| |D^{\gamma}g|\ , $$ in order to see that $\theta'=\theta''$ works if it is defined by $$ \theta'_{\beta}(x):=\frac{1}{\beta!} \sup_{\alpha\ge \beta} \sqrt{\prod_{i=1}^{d}(\alpha_i+1)!}\times\sqrt{|\theta_{\alpha}(x)|}\ . $$
Brief Feb 2020 addendum:
@Martin Sleziak: Thank you for the edit. I didn't know one could link to a specific page as you did for the reference to Horváth. That's great!
Request for references: I attribute these explicit seminorms to Horváth because I only saw them in the book I mentioned. If you are aware of an earlier reference where these seminorms appeared, please let me know.
The spaces $C^\infty_c(\mathbb R^d)$ and $C^\infty_c(\mathbb R^d)\times C^\infty_c(\mathbb R^d)$ are $LF$ spaces (inductive limit of Frechet spaces) and their standard topologies are not metrizable. We need only to check that for $K,L$ given compact subsets of $\mathbb R^d$, the restriction of $F$ to the Frechet space $C^\infty_K(\mathbb R^d)\times C^\infty_L(\mathbb R^d)$ is continuous (here $C^\infty_K(\mathbb R^d)$ stands for the $C^\infty_c(\mathbb R^d)$ functions with support included in $K$). Calling $F_{K,L}$ this restriction (valued in $C^\infty_{K\cap L}(\mathbb R^d)$), it is enough to check its sequential continuity, which is what you have already done.
The problem is solved here: https://math.stackexchange.com/a/1710723/41494
The solution given by the user Vobo is the following:
Let $B_n$ be the ball with radius $n$, $K_n=C_c^\infty(B_n)$ with its metrizable topology, $\varphi_n\in K_n$ a function with support contained in $B_{n}$ and $\varphi_n(x)=1$ for $x\in B_{n-1}$. First observe that $$ F_n\colon K_n\times K_n \to K_n $$ is a continuous map, which can be easily seen by the defining seminorms for these metric spaces.
Now let $U$ be a convex neighbourhood of $0$, i.e. $U\cap K_n$ is a convex neighbourhood of $0$ in $K_n$ for each $n$. Inductively for each $n$, you can find a $0$-neighbourhood $V_n$ of $K_n$ such that $$ F[V_n,V_n] \subseteq U\cap K_n $$ (by the continuity of $F_n$) and $$ \varphi_k V_n \subseteq V_k\,\,\,\,\, (1\leq k < n).$$ Set $W_n:=V_n\cap K_{n-1}$ and $W$ as the convex hull of $\bigcup_n W_n$. Observe that for each $n$, $W_n$ is neigbourhood of $0$ in $K_{n-1}$, so $W\cap K_{n-1}\supseteq W_n$ is one too, hence $W$ is a neighbourhood of $0$ in $C_c^\infty(\mathbb{R}^d)$. Now $F[W,W]\subseteq U$ would establish the continuity of $F$.
Let $\psi, \chi\in W$, i.e. $\psi=\alpha_1\psi_1+\cdots + \alpha_m\psi_m$ and $\chi=\beta_1 \chi_1 + \cdots + \beta_m \chi_m$ with $\alpha_i, \beta_i\geq 0$, $\sum \alpha_i = \sum \beta_i =1$ and $\psi_i,\chi_i\in V_i$. As $$ F(\psi,\chi)=\psi\cdot \chi = \sum_{i,j} \alpha_i\beta_j \cdot \psi_i\chi_j $$ and $\sum_{i,j} \alpha_i\beta_j = 1$, it it sufficient to verify $\psi_i\chi_j\in U$. Now if $i=j$, $$ \psi_i\chi_i = F(\psi_i,\chi_i)\in F[V_i,V_i]\subseteq U\cap K_i \subseteq U.$$ If $i\neq j$, e.g. $i<j$, then $\psi_i\in V_i$ and $\chi_j\in V_j$ and so $$ \psi_i\chi_j = (\psi_i \varphi_i) \chi_j =\psi_i (\varphi_i\chi_j)\in V_i\cdot V_i \subseteq U\cap K_i \subseteq U.$$