You are right in the first part: Science can only study the spiritual as it affects the physical. Thus, we have brain scans of those having "mystical" experiences or trances.
I don't see anything particularly "mystical" about them and neither does science. Unless, of course, by mystical you mean that we don't have a complete picture yet, but that can be said about just about anything.
But you are entirely wrong in stating that science assumes something does not exist if there is no positive evidence that it does exist. The correct response is that science remains agnostic: it does not make a statement either for or against existence.
That is false. Unless one has positive evidence that something exists, either directly or indirectly, it is for all scientific purposes assumed that such a thing does not exist. A good example would be the concept of Dark Matter and Dark Energy. For a long time we had no evidence at all with regards to these concepts, but since we now have indirect evidence, it is assumed that they exist even if no-one has ever seen or directly measured either. Of course, no scientist claims to know exactly what Dark Matter or Dark Energy is. The names are merely place holders until we get more information. But the point is that science only assumes the existence of things we have positive evidence for, either directly, or indirectly as extrapolated from other evidence.
To make the strong claim that X does not exist when there is no evidence is rather presumptuous.
You misunderstood me. Science does not make the claim that X does not exist. Why would it bother? Also, considering that proving a negative is logically impossible, making such a claim would in itself be rather unscientific. But things we don't have evidence FOR are assumed to not exist until we do.
Perhaps our instruments are merely not senstive enough, or we haven't looked in a particular area, or we haven't yet combined current data in a novel way.
This is, of course, possible, but until that happens, it is assumed that no such thing exists. You won't get a grant to investigate the mating habits of faeries for instance, because in a scientific sense they are assumed not to exist. But, as soon as someone brings in that first specimen/fossil/whatever that indicates that faeries exist... Well, that is a different ballgame.
Some of the greatest discoveries in sciences were things that were previously undetected: the neutron, cosmic background radiation, blackholes, etc.
Actually the cosmic background radiation was calculated (as in mathematically extrapolated from existing evidence) to exist, but it is true that the team that found it (Penzias/Wilson) discovered it by accident as they were trying to remove it as a disturbance to the experiment they were actually doing. It would have been discovered anyway though, seeing as Wilkinson and Roll were also in the process of constructing a radio telescope for just that purpose. Other than that you are correct.
Should science really say "We have no current data that a neutral particle of an atom exists, therefore, it does not exist."? Or does it/should it rather say "We have no current data that a neutral particle of an atom exists, therefore, we cannot comment any further on it at this time.?"
If science has no evidence, directly or indirectly, in favour of something existing it is in a scientific sense assumed that it does not. As explained above, that doesn't mean that one makes the claim that it -doesn't- exist. It is merely left out entirely.
Look, it works kinda like being an atheist. I cannot make the claim that there is no god at all, because that would be logically impossible for me to prove. However, since, at this time, there exists absolutely no evidence that there is a god I live in the assumption that there is no such thing. An atheist is NOT someone who denies the existence of god(s) but rather someone who does not believe that there are any.
So in that respect you could say that science is "atheistic" about anything we do not have evidence for.
The most science can say is that "At this time, we have no data/confirmation/etc". It cannot add the "therefore, it doesn't exist" because that would be a statement that has not been "proven" (in the way that science, using the scientific method, ever "proves" something.)
Proof is only used in mathematics. In the other scientific disciplines we use evidence.
I don't know where you are getting this stuff. Yes, correct, science cannot deal with something that is not falsifiable. That does not mean that science just waves its hand and says "Hey you, you unfalsifiable hypothesis, you don't exist." It simply means that that hypothesis is outside the purview of science: science cannot intelligently comment upon it.
For example, I find panoramic mountain vistas to be beautiful. Science cannot objectively verify the beauty of my mountains since beauty is by nature a subjective concept. Does this mean that my sense of beauty does not exist? Would science say that the concept of "beauty" does not exist?
We can empirically show that beauty exists, and by the use of an fMRI machine and some extensive testing we might be able to define it more accurately in the future. Or as Dawkins would have put it: "We're working on it".
You seem to be taking certain sound concepts about science and then taking them further than they are able to go.
Hopefully my clarification above will show you that this is not the case.