It's basically the hottest plasma we can make suspended in a magnetic donut surrounding by near absolute zero temperatures.
Certainly not in the conventional sense (although perhaps there's another way that helium might act similarly to a GHG that I'm not aware of).
Greenhouse gasses are molecules, i.e. multiple atoms bonded together. Those molecules can absorb photons of infrared light, which cause them to vibrate (as if the atoms were held together by springs). After some time, the vibration stops and an infrared photon is emitted.
The problem is: those photons are emitted in a random direction, unrelated to the photon that was absorbed. Half the time they will go roughly upwards, half the time they'll go downwards.
A photon of visible light (from the Sun) can travel down through the atmosphere without interacting much with the greenhouse gasses, since it has too much energy to be absorbed. This visible photon can be absorbed by other materials at ground level, e.g. by a plant, and its energy will eventually result in around 20 lower-energy infrared photons being emitted back up (on average).
These 20 infrared photons are readily absorbed by the greenhouse gasses, and each time they're absorbed, they get re-emitted in a random direction: half the time heading upwards again, but half the time heading back to the ground. This is how energy gets "trapped" by greenhouse gasses.
Helium is almost completely unreactive: it doesn't form molecules in the atmosphere, it just bounces around as individual atoms. Without bonds to vibrate (or asymmetries to spin), the only way it can absorb energy is by speeding up, and even this isn't very effective since its mass is so low. Fast-moving helium is also more likely to escape the Earth's gravity completely.
The frequency band for absorption and emission might be different, but the helium could absorb a shorter wavelength, then release a longer wavelength (not infrared) that can be absorbed by something else, then emitted yet again to be infrared
The lower the temperature, the higher your critical current is for a given superconductor. The reason HTS are a big deal isn't that you can have a superconductor at liquid nitrogen temperatures, but that you can have a very high field at liquid helium temperatures.
We still have plenty of helium. It's a byproduct of oil extraction and is still often vented because it would not be profitable of capturing it.
Also, per unit volume, the sun produces about as much power as a compost pile.
In fusion on earth, we want to be considerably more efficient than the fusion process in the sun, as we don't have as much space to work with. ITER is already a pretty big machine.