Beyond theory, are any organizations doing experimental work on Mars terraforming? In laboratory environments.
What do you mean "beyond research" ? Such studies are done in laboratories, or they are just games. You have a few "societies", such as the Mars Society, where people try to advance in such studies. Farming is a good subject, quite an easy one, lost cost, still important, and can eventualy not require huge instrumentation. Therefore, it has focussed some attention and still keeps going on.
At a less serious level, last year, I proposed such an experiment to young students (14-15) that were interested to do some experiments. They proceeded at one, about resistance to an important daily variation of temperature. On Mars the difference between night and day is about 70 to 80°C. We did a few "studies" between sub-0° at night and about 25 to 30°C at day, starting with different type of seeds, and chosing a comparative approach with witnesses. No major discovery ;-) Apart a cute intro for these young student to science reasoning :-).
At least one research group has conducted a chemical "experiment" simulating a possible greenhouse effect on Mars:
The data received by the Mars rovers suggests that Martian soil might be extremely acidic. If that was true at a large scale, then melting Mars' frozen water resources might be enough to cause both global warming and an increase of atmospheric pressure: if you put water on acidic minerals, greenhouse gases are released.
You can see the "experiment" at the time of the linked video:
I put "experiment" in quotes because this is very elementary. But this is, in my opinion, rather in favor of this concept of terraforming than against it. (in contrast to some of the far-fetched theories)
If you watch the whole thing, you can also see other plausible ideas and practical research. Though none of it looks as fancy as “domes“ or involves genetically Mars-tailored organisms.
This is by far not a comprehensive summary on everything that's going on, but you only asked if there are "any" organizations doing such work.
Yes, we are doing it now. It's called climate change.
By pumping the atmosphere full of ancient carbon we are terraforming the earth.
For those climate change deniers...
CO2 produced from burning fossil fuels or burning forests has quite a different isotopic composition from CO2 in the atmosphere. This is because plants have a preference for the lighter isotopes (12C vs. 13C); thus they have lower 13C/12C ratios. Since fossil fuels are ultimately derived from ancient plants, plants and fossil fuels all have roughly the same 13C/12C ratio – about 2% lower than that of the atmosphere. As CO2 from these materials is released into, and mixes with, the atmosphere, the average 13C/12C ratio of the atmosphere decreases.
This can actually have a noticeable effect on carbon dating as well.
The Suess effect
So in short, we are already experimenting with "terraforming" on our own planet.
Fossil fuels such as coal and oil are made primarily of plant material that was deposited millions of years ago. This period of time equates to thousands of half-lives of 14C, so essentially all of the 14C in fossil fuels has decayed. Fossil fuels also are depleted in 13C relative to the atmosphere, because they were originally formed from living organisms. Therefore, the carbon from fossil fuels that is returned to the atmosphere through combustion is depleted in both 13C and 14C compared to atmospheric carbon dioxide.