How can we show that the speed of light is really constant in all reference frames? I had a debate with a friend who cannot believe that the speed of light is constant.
He said something like: so what if in the Michelson-experiment the moving apparatus simply added a constant velocity to those photons. So even if you are on a train, if you throw a ball two perpendicular directions at the same speed towards walls that have the same distance from you, they will bounce back and get back to you at the same so this experiment doesn't really prove that the speed of light is really constant at any reference frame per se. 
Although I understand special relativity. I find his point quite hard to refute. Especially since in quantum mechanics we saw that light is made of particles again not waves. So I'm inconfident if I can begin with EM waves relative to the lab frame.
Are there other experiments and observations that confirm $c$ is really constant in all reference frames and there is no "slow" and "fast" light? So he who haven't really got the point of SR can see it?
 A: You find this hard to refute because your friend is correct in one sense: the MM experiment did not prove Einstein's second postulate of relativity, namely that the speed of light is constant for all inertial observers. 
Recall that the Michelson-Morley experiment was designed to detect motion relative to an aether, or material medium for light. If your experiment on an open train carriage measured the speed of sound, then you would indeed measure different speeds along and across the carriage. So the MM experiment cast serious doubt on the notion of an aether.
Now, it was well known that Maxwell's equations did not keep their form under Galilean transformations between inertial frames. This was thought to be fine because the notion of a medium for light was believed before the MM experiment, so that the wave equation for light should transform in the same way as the wave equation for sound between inertial frames.
So along comes Einstein and says, given there's no medium, let's see what happens to our physics if we assume that Maxwell's equations keep their form under a transformation between inertial frames. He postulated therefore that the speed of light would be measured to be the same for all inertial observers and concluded that (1) the transformation group was the Lorentz, not the Galilean group and (2) the time measured between two events would in general depend on the observer. (1) was already know at the time of Einstein's 1905 paper, (2) was radical.
So the MM experiment motivated the assumed Lorentz covariance of Maxwell's equations and thus the new relativity postulate that the speed of light would be measured to be the same by all inertial observers.
The second relativity postulate therefore comes into play only when we compare the light speed measured by different inertial observers. Someone observing a light source on your train would notice a very different transformation law from the approximate Galilean transformation law that would describe the ping-pong ball velocity transformation to an excellent approximation.
A: The easiest way would be to note that the speed of light can be measured by various apparatus, and that it is always measured to be the same (taking medium into account) regardless of its path relative to us.
In vacuum, light has never been measured to be moving faster or lesser than $c$, regardless of whether the Earth was moving towards, away, or perpendicular to it.
http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
http://en.wikibooks.org/wiki/Special_Relativity/Aether
A: Regarding the experiment mentioned with Francois Arago in 1810 measuring the speed of light when it hit the telescope, we are only measuring the speed of light once it hits earth's atmosphere. This does not tell us the speed of light out in space.
A: To test Lorentz invariance rigorously, one has to consider theoretical models where Lorentz invariance is violated that are not already ruled out. One can do that by considering the Standard Model and then adding terms that violate Lorentz invariance and studying the most general such model that is physically plausible.  This has been done in this article where new predictions for experimental signatures of Lorentz invariance violations were made, such as:


*

*Quantitative predictions on vacuum Cherenkov radiation

*Decay of a high energy photon into a electron positron pair

*Decay of high energy muons into electrons and photons

*Stable high energy neutral pions due to the decay to two photons becoming kinematically forbidden

*Stable high energy neutrons while protons become unstable at high energies
