Of course they do, and the engine uses smoothing groups as you set them in your modeling software. And you can use both B&W and colored speculars for certain effects. 2048 is a reasonable limit, the real limit is probably 8192, as in other engines of this era, but I didn't need to try that high yet. You can have more complex models, and all the tricks like baking averaged normals for unwrapped uvs work as they should. You can basically port models, textures, and materials between idtech4 and UE3/UDK, and idtech4 can have most of the UE3 material effects.
The only problem with textures and materials, as with all non-pbr engines, is that they're relative to lighting, so you have to either experiment with TDM stock lights or assume your own model (I prefer the latter).
That's actually amazing, gives me more chances to easily work around with, still sad there's no PBR for it though, but I might be able to code a material shader based on Phong (Beckman for example) for once.
For the lights, I'd rather do both, in another game I found an issue in a map because of the new shader, I only needed a tiny change in it that didn't change anything except for that one specific part.
IMO The textures are still holding up nicely, but it would be cool if whoever made props like the vases could spit out higher poly versions of the models and then just plop them into the game. (don't know if it is actually that easy)
With OpenGL3.0+ it would be possible, by use of "Tessellation" I'm sure, no need to remake a model so it's higher poly.
NOTE: The video provided is an example, a bad one though, because that whole engine seems to have been made to over-emphasize on tessellation, stairs don't need tessellation to not look like a "slide".
And of course, some cards don't support it, we have old timers here after all.