It's the high-frequency electromagnetic waves coming from the plasma tube that are doing the trick. And yes, you can in principle broadcast power using it, though it would be very wasteful. Still, no less a person than Nikola Tesla, to whom we owe credit for the majority of our modern power infrastructure and radio equipment (some sad stories there), was able to convince the venture capitalists of his era make a large-scale and very serious attempt at what you just suggested in the Wardenclyffe Tower. Alas, it was never completed and was torn down; only the foundations remain.
The effect of an electromagnetic wave on matter depends profoundly on how tightly bound the electrons are in a particular substance. In the case of metals, the conduction electrons are already delocalized and respond strongly to even the faintest whiff of the electric component of an electromagnetic wave, even at low frequencies and amplitudes. That's a good thing, too, since otherwise the whole discipline of radio broadcast and reception would not exist!
On the other extreme are compounds like rubber and most plastics, in which most of the electrons are very tightly bound and do not come lose easily at all. Radio waves have no noticeable effect on those until you reach intensities (e.g. microwave ovens, and even then most plastics are immune) where begin to heat or destroy the material itself.
Humans and other assorted bags of ionic fluids are an odd case. Salt water does conduct, but most of the conduction takes place via large and heavy ions, rather than super light electrons. So, radio waves do affect us, but the ions move so little in most cases that there's no real signal above background heat vibrations. In contrast to plastics, however, this mechanism makes us very much susceptible to microwave oven levels of radio waves, which are powerful enough to cause strong heating via those vibrations.
Fluorescent compounds are another curious case, but in a quite different ways. They are mostly akin to plastics in that the majority of their electrons are very tightly bound.
However, in a rough analogy to semiconductors, fluorescent compounds have a subset of electrons that are just loose enough to go sailing from certain charge centers if they are moderately provoked to do so. It takes a lot more energy than the trivial amount needed for metals, but also a lot less than the damaging energies of a microwave oven. A wave of the right frequency can repeatedly bump these electrons along, a bit like someone pushing a swing, until they can fully leave the charge center from which they originated.
Now, since these materials are more like plastics than metals, that presents a bit of a problem for the electrons, since they have no trivial path "just follow the charge gradient!" path back to the charge center, as would happen in a metal. So they can get a bit lost, for times that extend from microseconds up through minutes and even hours, maintaining substantial energy very much as though they were tiny one-electron capacitors in a broad, slightly positive background with lots of barriers in their way.
Eventually, however, they find a way back to one of the charge centers and fall into it with a bit of a thunk, recombining with a specific atom or molecule and releasing at that moment the sum of the energy they acquired during their outbound trek across the material. It's a bit like carrying a bowling ball up a gradual slope, then randomly finding a cliff edge to drop it back to the original level. None of your individual footsteps are powerful enough to drop the ball that hard, but cumulatively, they can add enough energy to create quite a bang if it is all released at once.
(I should note here for completeness that in other forms of fluorescence, such as a white shirt glowing under a black light, don't bother with any of this slow-climb business. An ultraviolet photon is more like a cannon that blasts the cannonball straight up to the top of the cliff and beyond, where it then falls back with some overall loss of energy.)
So, what is happening with your fluorescent bulbs in the vicinity of that plasma sphere is that the high-frequency electromagnetic waves are just strong enough to free some of the weekly held electrons from the charge centers and start them wandering around their mostly-insulating surroundings, accumulating enough total energy to give off a visible photon as they do so. The pigments in fluorescent lights are specifically chosen to allow fairly easy recombination, e.g. within a few milliseconds. However, even those materials retain a certain percentage of electrons that manage to get totally hosed and not find their ways back to positive charge centers for several minutes. It's easy to demonstrate: Go into a closet with with a fluorescent and let your eyes get used to the dark. Next, cover your eyes very well and turn on the light, just for a second or two. Turn off the light and look at it. You will see it still glowing, fading rapidly. That's due to the wandering electrons finding their ways back with a certain half-life. The effect can go on for some time, fading but never quite disappearing.
Best Answer
Background
Generally speaking, in plasma physics one uses the terms cold and hot to refer to the ratio of the thermal, $V_{T,j}$, to bulk flow speeds, $V_{o,j}$. Meaning, a cold(hot) plasma has $V_{T,j} \ll V_{o,j}$($V_{T,j} \gg V_{o,j}$). A quick reference of the possible thermal speeds can be found in this answer. For more information on cold vs. hot, see this answer.
For a proton-electron plasma, if $T_{e} \approx T_{p}$, then due to $m_{e} \ll m_{p}$ one can see that $V_{T,e} \gg V_{T,p}$.
To think about temperature in a plasma you need to be careful about the type of plasma. In a plasma that is not controlled by Coulomb collisions, the temperature is not as clearly defined as one might recall from fluid dynamics. One can still calculate the various velocity moments of the distribution function, but temperature becomes more of a measure of the average kinetic energy of a species in its bulk flow rest frame than the standard thermodynamic quantity.
Questions
Generally speaking, for a plasma to be considered "thermal" in the context of which you speak, the gas would need to be dominated/controlled by Coulomb collisions. The result would be a Maxwellian-like velocity distribution (i.e., gaussian).
The lower atmosphere (i.e., low chromosphere and photosphere) of the sun is thought to be a collisional plasma, thus much of the plasma is thermalized. However, once the plasma reaches the upper atmosphere (i.e., corona), Coulomb collisions start to lose relevance to other faster mechanisms like wave-particle interactions. This is all ignoring the complications introduced by transient phenomena like solar flares and coronal mass ejections.
The solar wind is generally considered to be weakly collisional at best, but for most scenarios it is usually assumed to be collisionless. The velocity distributions of both the ions and electrons are typically composed of several components.
Ions
For the ions, they are often composed of the following components:
Electrons
For the electrons, they are often composed of the following components:
Answer
So both the ions and electrons have multiple components, many of which are not modeled as thermalized distributions (i.e., non-Maxwellian profiles). Thus, stars are fully capable of generating what many people would call nonthermal plasmas.
This depends upon how the plasmas were made and what the relevant parameters (e.g., plasma beta, ratio of cyclotron-to-plasma frequency, etc.). It may also depend upon the type of ions used, since the mass ratio between ions and electrons is relevant for energy exchange processes between the two species.
The reasons relate to how the plasma is heated once formed. One way of doing this in fusion research is to use the instabilities that naturally occur within all plasmas. An instability is a mechanism by which free energy can be transferred to/from the electromagnetic fields from/to the particles. Generally, the free energy exists in the particles as a non-Maxwellian feature (e.g., a secondary beam or temperature anisotropy) and the instability then radiates an electromagnetic wave. During the process of the radiation, the energy used to produce the wave is taken out of the particle distributions in an attempt to remove the free energy. Once the wave exists, it can then further interact with the plasma (interesting side note: the wave interacts most strongly and efficiently with the types of particle distributions that radiated them in the first place).
Some of these instabilities are very good at transferring energy between the ions and electrons (e.g., lower-hybrid drift, ion-acoustic, whistler-types, etc.). These types of instabilities can lead to situations where $T_{i} \approx T_{e}$ (However, as I stated above, if the plasma is not controlled by collisions, then the temperature is really defined by component.). There is something to note though. The waves radiated by instabilities will interact with particles over a specific range of energies and phases, not the entire distribution. Thus, they often lead to further non-Maxwellian features rather than relaxing the distribution to an isotropic Maxwellian.
Answer
So to answer your question, I am inclined to think the temperature equilibriation in fusion plasmas results from the instabilities generated (and sometimes used) to heat the plasma. The plasmas in any lab device are generally created for a specific purpose, so the experimenters generally want specific parameters for various levels of control. Unfortunately, there are dozens of reasons why $T_{i} \approx T_{e}$ and why $T_{i} \neq T_{e}$ in plasmas. Without knowing further specifics, I cannot further elaborate.