The following is from Shaefer's "Topological Vector Spaces", sections 5.5 and 5.6.
For a locally convex (Hausdorff) $E$, the injection into its bidual is surjective iff every bounded set in $E$ is weakly-relatively-compact. In this case $E$ is said to be semi-refelexive. The map into the bidual need not be continuous. This bijection is continuous iff $E$ is also barreled. In this case this bijection is in fact an isomorphism of topological vector spaces and $E$ is said to be reflexive.
A general topological vector space which injects into its bidual, when taken with the weak topology, is clearly locally convex Hausdorff, so we can safely restrict ourselves to this class.
So, the answer to the first question is "the general class of locally convex Hausdorff topological vector spaces for which this is true is called semi-reflexive spaces" and the answer to the second is "yes, in particular it is true for reflexive spaces".