Do sequences fully specify the topology of $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$?

One does NOT understand $\mathscr D$ and $\mathscr D'$ by considering only sequences! For example (I think this was already noted in Hörmander's On the range of convolution operators (Ann. of Math. (2) 76, 1962, 148–170) it is a big difference for an injective operator $\mathscr D \to \mathscr D$ if it has a continuous invers (from its range back to $\mathscr D$) or only a sequentially continuous inverse.

As far as I remember V.B. Moscatelli worked on (perhaps even characterized) inductive limits of Frechet spaces such that every sequentiall closed subspace is closed. A nice article about the interplay of such general questions and analytical problems is Floret, K. (1980). Some aspects of the theory of locally convex inductive limits, Functional Analysis: Surveys and Recent Results II, (K.D. Bierstedt, B. Fuchssteiner, eds.), North Holland Math. Studies 38, Amsterdam, 205-237.