Since the first question doesn't seem to have been addressed directly in the answers so far, here are some suggestions. Firstly, since we are discussing topological vector spaces, I think the most natural question is to consider the forgetful function onto the category of topological spaces and since functional analysts are interested in function spaces, to completely regular spaces (of course, the case of sets can be incorporated by regarding a set as a discrete topological space). One then has a natural construction of the free locally convex space---one takes the free vector space generated by the topological space $X$ and provides it with the finest locally convex topology which agrees on $X$ with the original one. In our situation, this will be Hausdorff and will contain $X$ as a closed topological subspace. It is simple and natural to carry this one stage further and consider the completion of this space. It will have the corresonding univeral property, now for functions with values in a complete locally convex space. This space has a natural explicit representition, e.g., if we start with $[0,1]$, we get the space of Radon meassures on the interval. One of the nice things about this construction is that it can be varied almost infinitely and provides a unified approach to many spaces whose initial development was slow and painful---some of which are again forgotten lore.
As examples, we can consider spaces with the universal property for bounded functions and replace continuity by other smoothness conditions---uniform continuity if $X$ is a uniform space, $C^\infty$ if $X$ is an open subset of some euclidean space, holomorphicity (subsets of the complex plane or its higher dimensional analogues), measurablility if $X$ is a measure space and so on---similarly for functions on suitable manifolds. This provides a unifying approach to such topics as uniform measures, distributions, analytic functionals and so on.
As regards the second question, I have the feeling that functional analysts and topologists use the term compactly generated with different meanings. For the former, a locally convex space (in particular, a Banach space) is compactly generated if it contains a compact subset whose span is dense. For the latter, a topological space is compactly generated if it has the finest topology which agrees with itself on compact sets (otherwise known as a $k$-space or a Kelley space). As remarked above, metric spaces have the latter property and have the former one if they are separable. Further examples of spaces which have the latter property without being metrisable are the so-called Silva spaces, i.e. countable inductive limits of sequences of Banach spaces with compact interconnecting mappings. Many of the important spaces of distributions belong to this class, as do spaces of analytic functionals.