Is there an experiment that verifies the quantum state collapse is a physical process or a process of reality?

The two-slit experiment is a "classical" method that verifies collapse of the quantum state. This wave state collapse is a real process of nature (and therefore a process of reality), which results when a system in a superposition of other microscopic states interacts with a macroscopic measuring device. There are many ways of interpreting the how and why of such collapse, but physicists all agree that it is a process occurring in reality.

For example, see here for the Copenhagen interpretation, perhaps the most popular, and here for many other interpretations.

Now, having said that, your question on "physical process" is whether or not the process of collapse is physical in that are there dynamical equations that describe this collapse? The answer to that is no. You cannot really call "collapse" a dynamic process. This question is very philosophical. So once again I recommend you read the above interpretations.

This is a tricky one. First off, I will point out that the question you ask hinges on a philosophical point. The doctrine of scientific realism states that the things science measures are actually real things, part of reality. This doctrine says that things like atoms and photons are actually real in the truest sense of the word. Contrast this with instrumentalism, which argues that the things science talks about are just effective models which are good at creating accurate predictions of what happens in the physical world, but doesn't claim that they are real. An instrumentalist would happily predict the behavior of a system using photons as a tool, but would not take the additional step to claim that they are real.

Based on the wording of your question, I am assuming you are talking from the scientific realism perspective, in which we assume that if our scientific models always work, then the things they talk about are "real." In this sense, quantum mechanics is real. We have no experiments which show that it is wrong, and plenty of experiments which show that the simpler ways of thinking of things, like classical mechanics, are not correct.

I wrote an answer on this topic recently, regarding the meaning of the double slit experiment. The double-slit experiment is probably the most famous experiment demonstrating quantum mechanics, and the quantum eraser variants of them are the most demanding to explain without quantum mechanics. As it turns out, you can explain them using classical mechanics, but you have to resort to retrocausality, which is when a future event affects a past event. We generally don't like to think that way in physics, so we are much happier with a quantum explanation which is far more complex but is purely causal in the normal sense.

As for quantum state collapse, that is considered to be an interpretation, not a physical reality. Indeed, there are interpretations of QM which do not have state collapse. Quantum state collapse is a way of explaining what we see in QM if we assume that we and our observational tools are "classical" devices which can be explained solely via classical means, but the object under study (perhaps a nucleus) exhibits quantum behaviors. The interpretations predict what we will see, in a statistical sense.

The more "real" theory is decoherence. Theories built on quantum decoherence assume that we, and our measurement devices, are actually governed by quantum mechanics, but that the thing we are choosing to measure is something statistical, such as the expected average spin of 100,000,000,000 electrons. On these large scales, we find that the effect of "turning the crank" on the QM equations (aka "shut up and calculate") with a bunch of atoms that are in a "random" state yields results that get closer and closer to what classical mechanics claims occurs. And by closer, I mean so incredibly close to classical behaviors that we have never, and probably will never observe a large macroscopic object demonstrating quantum behaviors.

So QM is real in the sense of scientific realism. State collapse itself is an effect of the interpretation, but there is a real QM effect which occurs when dealing with large numbers of statistically distributed state dimensions which rapidly approaches what we call "collapse."

There is no dynamical process of collapse and there is no equation describing such a process. Wave function collapse is the consequence of certain interpretations of QM. I recommend the statistical interpretation, which avoids this concept.