"The gas then goes through a compressor, which turns it back into a liquid, but makes it hot in the process, so it goes through another coil that cools it with outside air before sending it back to expand in the first coil."
Not quite; The gas goes through the compressor and is compressed into a superheated vapour which enters the condenser and as it cools, becomes a liquid before sending it back to the evaporator.
(You can't compress a liquid)

It seems to me that if you take a solar water heater, an absorption refrigerator, and a Sterling cycle engine you could produce hot water, refrigeration, and electricity. Use the waste heat from the refrigerator's condenser to power the engine.
re; Mic
Not all liquids are incompressible.

Um no thats the refrigeration cycle this is reverse of that. It uses the properties of the refrigerant you don't actually compress the refrigerant. It uses heat to turn the refrigerant from a liquid to a vapor and that expansion powers the turbine. Then you cool the refrigerant back to a liquid state and use a liquid pump to push it back to be expanded and spin the compressor backwards. You need a temperature differential to make it work. This isn't new technology. This is used in waste heat recapture and geothermal applications. What makes this truly astounding and hats of to the STG team is they found a way to do it with off the shelf technology scroll compressors are everywhere. Before it required the turbine to be machined out of a single peice of aluminum. The problem is they leak. But a compressor holds the refrigerant better and reduces maintenance.

What's wrong with an off-the-shelf standard energy efficient heat pump such as the Mitsubishi Ecodan? We are installing loads of these combined with solar photovoltaics. Tried and tested technology with major backup. Can run on off-grid battery systems also. We have such a system operating in the Island of Sark, Channel Islands from an inverter.

re; Paul Fletcher CEO www.e-si.com
We can start with this is not an air conditioner it is an exotic steam engine driving a generator.
But then there is that you are using AC motors with a DC power supply.

We could call such a device a "heat pump."

why not just hot water panels and a storage tank? ? wle

Solar PV panels were first widely used in Papua New Guinea and not the USA or Europe. The cost of getting fuel to locations where power was provided by a generator was easily offset the one time cost of installing PV panels instead. A PV system producing electricity is as simple as anyone can imagine and takes no skills to install and requires no maintenance other than washing off the panels once a month.
The most efficient use of solar is to directly heat water and this can be done with a 100% passive system that requires no electricity. Leave it to MIT students to create a complex device that requires a computer to keep it operating and a power system to keep the computer running.
Want to see how this should be done read about the designs of Victor Papenek and his students. Start with his book, Design for the Real World.

Great idea but A/C scroll compressors only have a 3-1 expansion so this unit can't be very eff, no more than 15%.
If they designed a much higher expansion motor to extract more power of say 30%, then the other componants can be 50% of the size, lowering costs greatly.
A lower tech uniflow steam unit at 30% eff that they can fix, rebuild and maintain and even produce them would give not only energy but jobs too.

I commend the effort, goals and objectives in helping the developing world. But does it really help the developing world? To really help them and advance our own "developed world" why not deploy modern technology like Mr. Fletcher suggests as the the KISS rule applies in both developed and developING worlds.
Maybe I missed something but they didn't really point out where they acquired the mirrors, pipes motors, fans and all the other moving parts to make this apparatus work. A parabolic mirror can't be found under a rock so building one that is efficient, hauling it over rough terrain without damaging it and maintaining it for any length of time will cost a lot more work than to just use suitable technology. The probability of error increases as a square of the number of parts. (sorry)
My suggestion, in due consideration of the big picture, the most effective way to deploy heating and refrigeration to the developing world and to indigenous people might be to use the peltier system which is solid state and with no moving parts. The bill of materials for this project will fit on a bar napkin and can be hauled on the ground or frankly dropped by an aircraft to the "remote location" with relatively low low probability of damage. It and could be assembled with Swiss army knife.
One must consider all the parts and tools required to put it together, probability of a mistake in doing so, then where all this "stuff" will really come from and what happens if (when) a human is accidentally exposed to the "organic" refrigerant.
Why not try a small camping frig, 12V battery and PV panel or two. As demand mandates, scale that up to meet your requirement. The probability of it getting destroyed in route is much lower.
PS: Some indigenous people are that way by choice. Many have a higher quality of life than we in the "developed world" do.