I've never heard of something like this (nor have I managed to find a reference to it), but it seems like an interesting concept, and definitely not a short answer so I'll try to summarize it as best as I can.
The solid propellant is something like a dense rubber that once you ignite it through an ignition charge, will continue to burn until is finished. Unlike liquid engines where you can cut the propellant or electric-based propulsion where you can cut the power, there is no switch that can stop it, so my guess is that the reasoning behind whatever system you saw, is that by suddenly increasing the throat diameter, you will lose chamber pressure to some degree and the flames might get "sucked out" of the chamber effectively "turning off" the motor. I explain this because I'll base the rest of the answer based on this assumption.
I think is also important to mention that the fact that you have a schematic or a concept does not mean the design is actually manufacturable in real life (maybe a very low-temperature prototype in the lab just to test the concept), in fact, it's very common for an interesting concept to die once you start working what you actually need to build it.
For your first point, how did they design such a throat? My first guess is they don't. Chemical rockets have a combustion chamber at high pressure where the chemical reaction happens, then a motor/engine throat, where the gases product of that reaction are accelerated and reach their max speed, and then the nozzle, which manages the exhaust. Because the gases reach their max speed at the nozzle, that is also the point where the friction is higher and in consequence is the hotter part of the motor/engine. In solid motors that part is built using a material that resists really high temperature by slowly eroding through the whole burn. Liquid propellant engines on the other hand use complex liquid cooling systems.
There are limitations to a throat design you still need to respect when designing something like an adjustable throat. You still need to make sure the design does not generate additional friction or interrupt the gas flow since that would increase the heat even more and reduce the efficiency and thrust, so something made of moving parts and pieces is a very bad option, you want it to be smooth. You might consider maybe a hypothetical flexible material with a low thermal expansion rate that can be squeezed to reduce the cross-section of the throat, with an external mechanism to control the "squeezing", but if it's flexible enough to be squeezed, it would probably be deformed by the internal pressure of the combustion chamber and you also won't be able to contract the throat without some wrinkling of the material, which again would increase the friction. What I see is that there are a conflicting set of goals in the design of an adjustable throat that, without some fictional material, are not possible to meet at the same time.
Your second and third questions are simpler. Assuming the "turning off" mechanism works as I mentioned at the beginning, you won't really lose propellant during the "blow out" (maybe just the gases already in the chamber), it would be somewhat analogous to turning off a candle to relit it later, so for the third question purposes, the motor would be completely off, no flames, nothing.
Lastly, why is there noting out of R&D in this topic?
The non-existence of a material that can be used for this is a big reason, but even if there was a material that could be used you need to ask yourself if it's really worth it. In the end, like everything else in the space industry, it comes to tradeoffs. At this point is necessary to ask if it's worth designing, or if it's worth having at all.
You need to ask why would you want to have a restartable solid motor in the first place:
- Solid motors provide high thrust at low cost, but they are less efficient, so they are used mostly to get out of the atmosphere, either as the main booster like in the new Vega C Rocket (Vega-C) or as strap on boosters to assist the main stage like in the SLS (SLS). In non of those scenarios, you will be interested in turning a solid motor off to re-ignite it later.
- They are also used because are simple and reliable and can be stored with little maintenance for long periods of time. If you add mechanical components you would need to provide maintenance regularly and those will make the booster more expensive, heavier and less efficient. (Plus you would need to add an additional ignition charge for every time you want to turn it on again, which would also add mass).
- In a space scenario, for example in orbit changes or station keeping, you need very efficient engines, high thrust is not a priority and depending on the application it can actually be an issue. Is common to have engines or boosters where the thrust is not homogeneous, with small variations through the nozzle cross-section that over time could accumulate and throw the spacecraft off course if they are not accounted for. With a low thrust liquid propellant engine, these variations accumulate very slowly and you can use RCS systems to compensate, or spin the spacecraft along the engine axis so the variations average out over time. But with solid motors, the high thrust means these variations accumulate way faster and makes it very difficult to implement any corrective measure if required.
In general, I would say an adjustable throat is not worth it because it doesn't make a solid booster better:
- Adds complexity which is translated to more potential failure points and the need for additional maintenance.
- Adds mass which makes the solid motor less efficient.
- There is no scenario where being able to turn off and re-ignite a solid booster works better than current solutions.
- Not to mention there is no material that would allow its manufacturing.
