Experimental proof of space expansion We know the universe expands with the space expansion. This follows from the FLRW solution that nicely explains the observed Hubble flow. (Since any existing acceleration is extremely small, let's assume for simplicity it is zero.)
A uniform expansion of space does not apply any forces on bodies, so they move exactly the same way as they would in a non-expanding space. For example, if two bodies are at rest relative to each other, but space between them expands, then they would remain at rest relative to each other the same way as they would in a non-expanding space. In other words, the expansion of space cannot be detected by observing the motion of massive bodies (as long as they don't cross the cosmic horizon).
The expansion does affect the speed of light, as observed remotely. The expansion also creates a cosmic horizon, beyond which nothing can be detected. Have these effects ben observed? Are there any experimental results showing that space actually expands as opposed to the galaxies simply flying apart on inertia?
 A: You are not going to get an experimental proof of space expansion, because space expansion is just an interpretation, not an observable fact. If you want to describe cosmological expansion in terms of motion rather than expansion of space, you can. There is a fundamental ambiguity in the informal verbal interpretation of these cosmological metrics, because we want to visualize expansion using mental images of the Hubble flow, but those mental images implicitly assume that there is some way to define the velocity of galaxy A relative to distant galaxy B. GR does not offer any such definition. When we talk about distances between galaxies, or about recession velocities, people who understand GR realize that this is shorthand for a more complicated set of definitions that are to some extent arbitrary, and are not generically built in to GR. E.g., to talk about the "distance" between A and B, we have to define a 3-surface orthogonal to the Hubble flow, and then measure the metric distance along a spacelike geodesic inside that surface.

In other words, the expansion of space cannot be detected by observing the motion of massive bodies (as long as they don't cross the cosmic horizon).

Detecting a cosmic horizon doesn't have any special logical status here. There are all kinds of other observables, such as Doppler shifts, that rule out any attempt to interpret cosmological observations in terms of the motion of matter on a background of flat spacetime. You seem to be trying to make a dichotomy between spatial expansion and Newtonian gravity, but the observationally verifiable dichotomy is between GR and Newtonian gravity. Spatial expansion is just one way of talking about GR, and it's an optional way of talking about it, just as the Copenhagen interpretation is just one optional way of talking about quantum mechanics.
A: 
The expansion also creates a cosmic horizon, beyond which nothing can be detected. Have these effects ben observed?

We see photons which have been emitted beyond the Hubble sphere. These photons have been receding from us in the early universe. If you say "cosmic horizon" you probably mean the particle horizon, which is today 46 Glyr away from us. We can't detect photons from beyond this horizon per definition, as the particle horizon limits the observable universe. 

Are there any experimental results showing that space actually expands as opposed to the galaxies simply flying apart on inertia?

No, there aren't. Expansion of the universe means increasing distances between comoving objects. Whether space expands or distant galaxies are moving away from each other depends on the chosen coordinates. So neither of these views - being not invariant - may be considered as a measurable physical phenomenon. Space expands in FRW-coordinates and galaxies move away in normal-coordinates.  
