Lower and upper bounds on the diameter of 3-regular graphs obtained after reducing practical real world problem instances to #3-regular Vertex Cover
A lower bound of approximately $\log_2 n$ still follows for arbitrary 3-regular graphs, based simply on the fact that there are at most $3\times 2^i$ different paths of length $i$ from any  given starting vertex and one needs the diameter to be large enough to give a different path to each of the $n$ vertices. The random graph example that you mention (or the example of a complete binary tree with some extra edges thrown in to make it 3-regular) shows that this is tight to within a constant factor, and I'm pretty sure the actual lower bound is really $(1+o(1))\log_2 n$.

As for an upper bound: there exist 3-regular graphs with diameter $\Omega(n)$, for instance the Möbius ladders.

If you want to know the diameters of graphs coming from specific hardness constructions, you'd have to look at the details of those constructions, but my guess is that in most such cases it would be straightforward to tack on either a large binary tree or a large ladder in order to make the diameter as small or as large as you want without much changing the hardness of the instance.