You pretty much hit the 4 techniques popular for surfaces in the last few years. From my experience with trying to integrate these techniques into existing pipelines, the two worth your time are flat-dicing + displacement textures and Phong. The main reason is that they are low investment compared to the other two.

Like mentioned, flat-dicing + displacement textures gives pretty full control to the artists. Phong is useful for smoothing objects, specifically their silhouettes. One problem with Phong can be that it balloons objects. A common solution is to clamp how much it can displace vertices.

Another use I've seen is to tessellate particles so that you can do shading in the domain shader. It's somewhat of a happy medium between doing vertex shading versus pixel shading for lit particles.

As for tessellated hair, I feel it's a mixed bag. You leverage tessellation for hair to generate extra line segments and/or copies of the line segments of hair. They're fully parameterized so you can use that however you like to make them visually different. My experience was a big perf hit for a single object. It also was hard for an artist to work with since it was very new and they had to spend the majority of their time in engine with it. Artist like control and a parameterized system can be tough to get all the control they desire. I've also heard from others (see the literature from GDC 2013 for Tomb Raider) that just pre-tessellating hair offline was more efficient. This also gives more direct control back to the artist again.

As for the theoretical, The concept of a limit surface you mentioned is key. You'll likely want to look ahead into how content pipelines may change in the next decade if you really want to go into that area. Another area I see as related and worth mentioning is changes to content authoring and pipeline such as ptex. In the long run, Content tools, pipelines, and maybe even methods are going to have to change before tessellation can be integrated in a productive and tight manner imo.