When and how was it 'concluded' that there is no air in space? When was it confirmed/concluded that there is definitely not air in space? And how was the conclusion reached?
We see the discussion of 'ether' before Special Theory Of Relativity and there it seems evident that there was no air in space (everybody knew it), which is long before satellites were sent in space.
 A: The discovery that there should be no air in space perhaps goes back to when gravitational force was being formulated, and the understanding that it causes pressure gradients in the atmosphere, all in about the mid to late 17th century.
When scientists learned that gravity pulls down massive objects (and the atmosphere contains massive amounts of air), they concluded  that this results in an air pressure gradient$^1$, so that at some height that pressure would be (close to) zero.
During the same period, gravity was mathematically formulated by Newton, so that the idea that air tends to be closer to the surface of earth, and (specifically, calculations on how gravity would act on air columns) the height at (roughly) zero pressure was calculated. The strength of the gravitational force is $$g=\frac{GM}{r^2}$$ so that for values of $r$ closer to the earth's surface, the field is strongest and drops off as $r$ increases. Therefore air has maximal pressure close to the surface of the earth and becomes close to zero pressure (vacuum) at some point much higher$^2$.
You also seem to have alluded to the Michelson-Morley experiment, that was actually an experiment to determine the possible existence of an ether (medium for the propagation of light). This was unrelated to the above.
$^1$ A few years before Newton formulated his universal law of gravitation, it was understood that gravity causes a pressure gradient. In fact, Evangelista Torricelli designed the first barometer in 1643, confirming that such a pressure gradient due to gravity does indeed exist.
$^2$ See Kármán line which is defined as the "boundary" (the point at which air pressure approaches that for the near vacuum of space) between the earth's atmosphere and space which is about 100 kilometers (or $\approx 63$ miles) where the density of air approaches a small number, effectively a vacuum compared to the density at the surface of earth.
A: Long ago, experiments were performed to measure the variation of atmospheric pressure with height above the earth using kites (see: Eddy kite) and balloons. These measurements went all the way up to about 30,000 feet and showed that the pressure decreases exponentially with height, and by extrapolating the resulting graph to near zero pressure, it could be concluded that at a certain height above the earth, there was essentially no air. This meant that in space, there was no air at all- and that there was a vacuum instead.
Later measurements at higher altitudes using sounding rockets confirmed this result.
And as pointed out by Joseph H, this result was checked quite accurately from theory knowing how gravity would act on a column of compressible air. To do this required measurements of temperature as a function of height, which were also performed using balloons, kites, and later, rockets.
This has nothing to do with relativity and the (hypothetical) ether.
A: The motion of planets has been observed for hundreds or even thousands of years. Newton explained the motion of planets and moons as caused by gravitation and gravitation only, and managed to calculate their orbits with high precision. These orbits would not work the same way if there was even low pressure air or gases out there, slowing down the motion of celestial bodies.
