Yes. Thermo-mechanical behavior of materials is a common link.

Yes, of course! If you apply your studies to the functional and structural behavior of polymer based materials.

Think that plastics and composites are being largely used  in the automotive industry nowadays  and their increasing use will depend mostly on the development of new low cost resins and manufacturing processes.

For instance, in all industrial fields the strategy to substitute rationally metals parts with composites in light weight application or even ceramics for insulating purposes is a prerogative of Mechanical Engineers ( that depends of course on the level of knowledge in polymer science).

Polymers Science field includes rheology and viscoelasticity, the fundamental knowledge to tackle the problematics of manufacturing processes of polymer-based parts. This is only part of the story. Polymers (including polymer based composites) are "materials" like metals, ceramics or glasses, but their growth rate is still increasing while that of "traditional"materials is levelling off. So that, sooner or later you , as Mechanical Engineer , will have to do with polymers.

Much need for mechanical understanding of testing and mode of failure in polymers. The area grows as plastics displace metals in auto and aircraft manufacture.

Yes. 

Now a days due to increased demand and supply of novel materials in almost all aspects of life we do require systematic materials testing experts. Notably, more than 70 % of these materials in one or other form comprised of polymers. 

Definitely, a mechanical engineer has potential role to study and extend the scope of polymeric materials in current thrust. The mechanical property study and design of mould are very good subarea under mechanical engineer.

I am a degreed Mechanical Engineer for nearly 30 years, and a degreed Polymer Scientist for nearly 20 years performing work in this combined area my whole career. 

I estimate that there are very few of us; I am curious as to how many..

The scope can be fairly broad: Composite materials, biomaterials, aerospace materials, adhesives, elastomers, etc. Characterization method expertise is needed in viscoelasticity (thermoelasticity), fracture mechanics and multi-dimensional stress conditions. I do know from experience that these are areas that needs such multi-disciplinary backgrounds.

An area that I have been involved in is surfaces (esp. of biomaterials), where chemistry and physics principles 'collide' during adhesion, abhesion and prevention of adhesion. A combination of these disciplines have a lot to offer. 

It does not need to be too basic, too. Many times the initial thought is "mechanical testing of polymers". This is true, but can be too narrow.