Is there a characterization of the locally convex spaces with the property that they are sequentially dense in their completion. In other words, under which conditions on a locally convex space $E$ is every element of the completion of $E$ the limit of a Cauchy sequence in $E$?
Does this property already imply that $E$ is metrisable?
I don't think that your general question has a satisfying answer (in the sense of a characterization which is evaluable in concrete cases). The answer to the second question is negative for some LB-spaces (countable inductive limits of Banach spaces). Although it is an open question whether every sequentially complete LB-space is complete there are special classes of so-called Moscatelli type LB-spaces which are sequentially dense in their completion. J. Bonet and S. Dierolf worked a lot with such spaces (e.g., Bonet, José; Dierolf, Susanne, On LB-spaces of Moscatelli type. Doğa Mat. 13 (1989), no. 1, 9–33).