The real evidence of this is from the last experiment under "Sequential Experiments". Essentially, you only select particles with spin-up in the z direction, then you select particles with spin-up in the x direction. So the resulting particles should be up in both the z and x direction. But if you measure in the z direction again, you find an equal distribution in up and down, indicating that you can't simply treat the spin as both pointed in the x and z direction, and that measuring in the x direction has scrambled the previously well-defined z direction.

This experiment shows that measuring the spin direction in different axes does not commute. Measuring in one direction scrambles the other, which is equivalent to saying measuring in x then z is not the same as measuring in z then x. This fact is inherently related to the notion of a superposition. If a particle's spin direction is well-defined in one measurement basis, it is not well-defined in another - meaning it is in a superposition state in that measurement basis.

You might ask - why can't I describe the system after measuring in the x-direction as just a random mixture of up and down in the z-direction? Physicists use something called a density matrix to describe systems that have both some degree of quantum superposition and classical randomness. One way to measure the degree to which some stream of particles is a random mixture or not is to interfere particles in that stream with each other.

In the Stern-Gerlach experiment, after measuring in the x-direction, if the information in the z-direction was actually simply randomly scrambled, the probability that any two particles from the stream are truly identical is 1/2. If the particles are all identically in a superposition state, then any two particles will always be identical.

You can actually test the indistinguishability of two particles by doing an interference experiment. One very nice example of this is this experiment: https://arxiv.org/pdf/1312.7182.pdf Two atoms were trapped next to each other using lasers. If these atoms have the same spin, they're indistinguishable. If they have different spin, then they are distinguishable, and won't interfere with each other. In fig. 3, you can see varying levels of interference depending on how well-aligned the spins are.

I am not quite sure what you mean. There is tons of experimental evidence for quantum mechanics.

Specifically for electron spin, you seem to agree that it has spin 1/2 (because you say that it can be spin up or spin down). But that already concedes that there are superpositions of up and down. Spin left, spin right, spin straight forward, spin backward are all equal superpositions of spin up and spin down; and all these states are trivially constructed.

Super Position has real analogs in all wave based systems

https://en.wikipedia.org/wiki/Superposition_principle

That is reality. I mean if you are actually questioning the wave-particle duality of nature then yeah, you technically should.

Particles aren't real, they're just waves.