Do photons truly exist in a physical sense or are they just a useful concept like $i = \sqrt{-1}$? Reading about photons I hear different explanations like "elementary particle", "probability cloud", "energy quanta" and so forth. Since probably no one has ever seen a photon (if "seen" it supposedly - and rather conveniently - ceases to exist) but many experiments seem to verify its properties (or are they maybe adjusted to fit the experiment). I thus can't help wondering if the "photon" is then just a physical/mathematical tool with inexplicable properties (like zero mass - but affected by gravity fields - and constant speed c in space) invented to explain some otherwise unexplainable phenomem and to supplement the elementary particles and their interactions. In short: Are they real or imaginary!  Does anybody know? Or maybe the answer is "blowing in the wind" because to most physicist it probably just does'nt matter as long as it works (as the alternative healers say). Sorry if I seem a little sarcastic here and there.
 A: There is lots of experimental evidence that the electromagnetic field exchanges energy with atoms in discrete chunks, and if we call these chunks photons then photons exist. Which is all very well, but my guess is that you’re really interested to know if the photon exists as a little ball of light speeding through space at $c$, and if so then, well, that’s a complicated question.
Actually all particles are more elusive than you might think. Many of us will have started our journey into quantum mechanics with the wave equation of a free particle, and been surprised that the solution was a plane wave that didn’t look anything like a particle. Then the teacher tells us we can build a wave packet to make a particle but, well, this isn’t all that convincing. Making a particle by constructing a wave packet seems awfully arbitrary for objects that are supposed to be fundamental.
In fact non-relativistic quantum mechanics doesn’t tell us anything about why particles exist and where they come from. It isn’t until we get to quantum field theory that we get a reason why particles exist and an explanation for their properties, but even then particles turn out to be stranger things than we thought.
When you learn QFT you traditionally start out by quantising a scalar free field. If we do this we find the field states are Fock states, and we interpret these states as containing a well defined number of particles. Acting on the vacuum state with a creation operator adds a particle to a state, and likewise acting on a state with the annihilation operator removes a particle. All this may sound a bit abstract, but it actually gives us a concrete description of what particles are. The particle properties, like mass, spin, charge, etc, are properties of the quantum field, and all the particles are identical because they are all described by the same field. So the theory immediately tells us why e.g. all electrons are identical, and it describes how particles can be created and destroyed in colliders like the LHC. Right now quantum field theory is the definitive theory for describing what particles are and how they behave.
But these field modes that represent particles look awfully like the plane waves that we started with back when we first learned QM. So the particles described by QFT still don’t really resemble particles in the intuitive sense of a little ball. And worse is to come. Fock states only exist for the free field i.e. one in which particles don’t interact with each other. And that’s obviously a useless model for particles like electrons and photons that interact strongly. In an interacting theory the field states are’t Fock states, and they aren’t even superpositions of Fock states. In fact right now we don’t know what the states of an interacting field are. The best we can do is calculate their properties using a perturbative approach or a lattice approximation.
But let’s get back to photons. We don’t quantise the electromagnetic field because it isn’t manifestly Lorentz covariant, so instead we construct a field called the electromagnetic four-potential and quantise that. And now we have a definition of the photon in terms of the states of this field. As long as we are dealing with situations where interactions can be ignored we have a nice clean definition of a photon. And we can describe the creation of photons by adding energy to the modes described by the quantum field and annihilation of photons can take energy out of the modes and add it to e.g. a hydrogen atom. In this sense photons are real things that definitely exist.
But this photon doesn’t look like a little ball of light. In fact it doesn’t look like a light ray at all. Constructing a light ray involves taking a coherent state of photons in a way that I confess I don’t understand but I know is complicated. This is the domain of quantum optics and I wish you many happy hours attempting to learn it.
This is the point made in the paper by W. E. Lamb that I mentioned in a comment. There is a long and ignoble history of people imagining that light rays are just hails of photons, then getting confused as a result. The only time we really see light behaving as a photon is when it exchanges energy with something. So when an excited hydrogen atom decays a photon is emitted. Likewise a photon can be absorbed by an atom and excite it. As the light propagates to or from the atom it is rarely useful to describe it in terms of photons.
I feel like I’ve gone on and on at some length without really answering your question, but that’s because your question doesn’t really have an answer. QFT, specifically quantum electrodynamics, gives us a very, very precise description of what photons are and I suspect most of us would say that of course photons really exist. They just aren’t the simple objects that most people think.
A: 
Do photons truly exist in a physical sense or are they just a useful concept like i = sqrt(-1)

They truly exist in a physical sense. 

Reading about photons I hear different explanations like "elementary particle", "probability cloud", "energy quanta" and so forth. 

I'm sure everybody is familiar with the term "elementary particle", and that lots of people will be cool about "energy quanta", but for myself I've never heard anybody describing a photon as a "probability cloud". Can you give me a reference for that? 

Since probably no one has ever seen a photon 

In a way you're seeing a whole host of photons right now. So many that they build up a picture. And people have definitely detected photons in experiments. 

(if "seen" it supposedly - and rather conveniently - ceases to exist)

Not always. The photon has an E=hf or E=hc/λ wave nature. Think of detecting a photon as something like detecting a wobbling jelly with a big stick. It stops the jelly wobbling. But there is such a thing as weak measurement. Weak measurement is like using a toothpick instead of a big stick. There's also Compton scattering. The photon is "detected", but it doesn't cease to exist.  

but many experiments seem to verify its properties (or are they maybe adjusted to fit the experiment). I thus can't help wondering if the "photon" is then just a physical/mathematical tool with inexplicable properties (like zero mass - but affected by gravity fields - and constant speed c in space) invented to explain some otherwise unexplainable phenomema and to supplement the elementary particles and their interactions. In short: Are they real or imaginary! Does anybody know? Or maybe the answer is "blowing in the wind" because to most physicist it probably just doesn't matter as long as it works (as the alternative healers say). Sorry if I seem a little sarcastic here and there.

They're real. We can make electrons (and positrons) out of photons in pair production. And you are made out of electrons and other particles that are equally real. 
A: As you can guess from the answers which vary widely from "yes" to "no," your question touches on a very sensitive topic for those who follow science.  The answer will quickly boil down to frustratingly phrased questions like "what is real" because what you ask is sufficiently tricky.
In the world of philosophy, science is categorized as part of "empiricism."  Empiricism is the philosophy of what we can know using our senses (aka "empirical observations").  Empiricism is a subdiscipline of epistemology, the study of what we can know.  Epistemology is separate from ontology, the study of what is "real," so from a philosophical perspective science does not actually make statements about what is real and what is not.
When you look at the evidence for the "photon," what is provided is a large body of empirical observations done by scientists.  In each case, we find that if we model the world as though photons exist, the results of the experiment are consistent with the predictions of that model.  It doesn't say that photons exist, and it doesn't say that they don't exist.  It merely says that a model which declares that photons exist is effective at predicting the results of past experiments.
If you believe past results predict future performance, don't play the stock market, but science has shown that its models have an astonishingly good track record of predicting results.  I would argue its results are far better than any other system out there, although to make such a claim we would have to sit down and agree on a metric for comparing systems first.  If you need to predict what is going to happen in a system, modeling the system in a way that includes photons is more than likely to give you a solid answer.  Science prizes itself on its ability to make predictions no other system can make, and make good on those predictions.
Science has been so effective at this that we start to get lazy with our terminology.  We start to say "photons exist" or "photons are real."  At the philosophical level, this is called abduction.  Abduction is in the same category as deduction and induction; it's assuming that the most likely hypothesis is true.  We assume that, without any better hypothesis, photons must simply exist.  This is not part of the scientific method; it's not an empirical claim.  However, the models are so bloody good at predicting the future that the more lengthy phrase "the universe is well modeled with the assumption that photons are real" is just not worth the extra breath it took.
Unfortunately, this abductive step can get us in trouble.  You mentioned you've heard of photons as being described as "probability clouds."  This is because, as we peered deeper and deeper into the universe, we noticed that modeling things as photons stopped fully describing what we saw.  Strange experiments like the double-sit experiment started to suggest that we can't just model light as a bunch of individual photons.  Experiments like that one were designed intentionally to push at corner cases where the old models simply broke down.  We then replaced them with new models which are much better at predicting results.
Of course these new models need to line up with all of the old empirical evidence gathered before this new model was formed.  As we step away from the difficult corner-cases proposed by the double-slit experiment, we find that there was a strong connection between some of the probability distributions which arose from the quantum mechanics wave functions and the "photons" that were assumed in the old model.
This, to me, is one of the brilliant parts of science.  By making these connections between the models, we can say "as long as you stay away from these particular corner cases, you can get away with modeling light as photons, because the errors you pick up are small."  Think about how amazing that is.  We're comfortable enough with our models to say "even if you aren't using the most advanced model science has to offer, we can still safely use it and put limits on the errors."
Regardless, in quantum mechanics, at its deepest level, there are no "photons."  There are waveforms which are continuous through space and time.  However, in many many cases, the behavior of these waveforms is discrete enough that we can capture part of the waveform and say "this part is a photon."  But it's really our decision to call it a photon.  Empirically speaking, it's a good decision.  In well over 99% of cases, its a good enough decision to make predictions, and that's what we want from science.
So do photons exist?  Nobody knows.  Our best models of the subatomic world, those of quantum mechanics, all show behavior that's right in the place where photons should be.  We have no particular reason to assume they don't exist.  Whether that is enough for you is really a question of personal preference and philosophy.
A: Photons exist and best described as packets of energy or particles. Of course the term particle is always trivialized by the strawman description of a ball or ball of light. A particle is part of something else and it does not need to be a solid ball. In fact it's just another  system of its own. A photon could be described as an oscillating system of its own which propagates through space at the speed of light. An individual photon could be built upon a minimum of four principles.
(1) An accumulative mass mass too small to calculate at this time.
(2) A Direction and speed
(3) A frequency which could indicate a systematic oscillation of its own.
(4) and overall angular momentum of the system which could be perceived as polarity.
As the photon propagates through space these four principles play apart with the most important being the oscillating frequency. 
The energy from this internal oscillation contributes to the photons energy and the photo electric effect. 
The photons speed and mass have a little to do with it.
This is just one way to physically describe a photon without resorting to a sarcastic model of a ball.
A: All experiments need a theoretical framework within which they are interpreted. Raw data on its own tells you nothing. 
Photons (and other particles) are part of an extremely useful framework for understanding many different phenomena, from esoteric high energy collider experiments to things we experience every day such as the blackbody spectrum. So, I use that language to understand and interpret a wide class of experiments.
However, at the same time, we can never say our current theory is definitely the correct way to look at things. There is always a chance that future experiments, or insights, will lead us to understand that our current theory is simply an approximation to a different theory, or that it is one of several equivalent representations for the same physics.
If someone were to come along tomorrow and propose an alternative way of looking at things in which photons did not appear explicitly, but which did explain all observed phenomena adequately, I would have no objections to using that language. (To some extent I could say that quantum field theory allows you to do this by interpreting scatting cross sections as a certain projection of n-point correlation functions of a quantum field). Indeed, I even could be convinced to prefer the new theory, if the new theory was able to explain new phenomena that the old theory could not explain. 
However, this new theory would still need to account for many of the observations we associate with the existence of photons--such as the clicks in a photomultiplier tube. It could well be the case that the new theory gives a less intuitive explanation of these phenomena. Then I would still prefer to think of photons as being a useful approximation--they capture enough of what is going on that I would say they are a good description in an appropriate limit even if they are not fundamental objects in the new theory.
So, do photons exist? It's hard to say no, since they provide such an elegant explanation of so many observed facts. But, I can't unequivocally say yes, since it is always possible there are other ways of explaining the same data, and it is possible that some of these other theories lead to a deeper understanding of physics.
PS -- this is a side comment, but I don't accept the premise that $i$ doesn't exist (for whatever definition of 'exist' you use for mathematical objects)... if you accept the existence of irrational numbers, then $i$ is really not stranger. 
