Sequential and topological duals of test function spaces

A first general remark: A topological vector space is metrizable, if and only if it is first countable. To answer your questions:

• Yes, Q1 and Q2 are true. Let $K_i$ be a defining sequence of compact sets for the space $\mathcal{D}$. As the topology on $\mathcal{D}$ is the final topology (i.e. the finest such that all injections $\mathcal{D}_{K_i} \to \mathcal{D}$ are continuous), a map $T$ on $\mathcal{D}$ is continuous, iff its restriction to each $\mathcal{D}_{K_i}$ is continuous. And for this, sequential continuity of $T$ is sufficient (assuming the known fact, that a sequence of test functions converges iff the support of the functions are contained in one $K_i$ and the sequence converges in $\mathcal{D}_{K_i}$.
• Q3 and Q4: Don't know if there exist relevant examples. I would always require continuity.