Did quantum fluctuations create matter and energy out of nothing? I found a popular science article explaining that the existence of the Higgs particle can be interpreted to mean that matter and energy are created by quantum fluctuations of nothingness (i.e. the vacuum).
Is this a sensible interpretation?
An answer in the affirmative could, for example, come by showing that a reasonable interpretation of the mathematics shows mass/energy created from vacuum.
 A: The question of how precisely matter and energy arose in the universe in the first place - "what really happened at the Big Bang" - is unsolved. We don't know what exactly happened, and that article took a significant achievement, a much improved a priori prediction of hadronic masses from QCD lattice simulations, and made it sound like something else entirely.
The problem is that "quantum fluctuations of the vacuum" doesn't actually mean anything like those articles make it sound like. If you look at In layman's terms, what is a quantum fluctuation?, the only rigorous meaning we can give to a "fluctuation" is that we have some average expectation value in the vacuum but the actual measurements can fluctuate around that. It's completely unclear how such a non-zero standard deviation should be related to "creation of mass". The Higgs field gives other particles mass by having a non-zero expectation value, not by fluctuating around that - in most states, there is some fluctuation, but that doesn't mean anything special on its own. For more on the Higgs mechanism, see How does the Higgs mechanism work?.
For more on the contentious topic of creation ex nihilo, see for example the following questions: How can space and time arise from nothing?, Stephen Hawking says universe can create itself from nothing, but how exactly?, Is it possible all matter in the universe emerges from nothing?, Does science show that matter and the universe were created out of nothing?
A: The question of why things exist trips up a lot of people (including physicists), but the words "vacuum" and "nothing" refer to very different things. When a physicist talks about the vacuum, they are talking about a region of space where there are no particles. But, in modern theories of fundamental particles, a particle is a stable excitation of an underlying field. There is an electron field, a photon field, a Higgs field, among many others. Each of these spans the universe. Ripples in these fields are experienced by us as particles.
You can think of a field like the surface of a pond. A smooth surface corresponds to the vacuum, where nothing happens (no particles). Toss a stone into the pond, and you excite fluctuations. The stable ripples that travel from one side of the pond to the other are the particles that we detect in our experiments. You can have a pond without ripples, but not ripples without a pond. In the same way, there can be fields without particles, but not particles without fields.
Now, random fluctuations in a field can give rise to particles. But, is this creation from Nothing? We already have several existent things: space, time, and fields. Otherwise, what would be fluctuating? Space, time, and the fields seem like a lot of Something to me, and (depending on how you define Thing) the laws of physics exist as well. So, no. As far as we can tell, nothing comes from nothing. Physicists are completely in the dark as to what actually happened at the $t=0$ moment of the Big Bang. Our theories and equations blow up before they reach that moment.We have a pretty good understanding of everything that happened 1 second after the Big Bang, so we're not far off.
The physicist Lawrence Krauss wrote a book a few years ago called "A Universe From Nothing" that set out to show exactly how a universe could spring from the void--from Nothing. Unfortunately, he fell into exactly the vacuum vs. nothing trap that I mentioned in the beginning. The philosopher David Albert wrote a harsh but fair rebuke in his New York Times review.
It may also be that the state of existence where nothing exists is impossible, if not logically incoherent, but this is pure speculation at this point.
A: 
Did quantum fluctuations create matter and energy out of nothing?

No. Matter can be created from gamma photons in pair production. Those gamma photons are not quantum fluctuations. And we know of no way to create energy.  

Has it ever been experimentally confirmed that matter and energy were created and are created by quantum fluctuations out of nothing all the time? 

No.

I thought it was just a hypothesis until I bumped into this link which claims that it has been confirmed but I'm not really sure. What is the status of this claim?

It's popscience misinformation I'm afraid. I'll take you through it:

Matter is built on flaky foundations. Physicists have now confirmed that the apparently substantial stuff is actually no more than fluctuations in the quantum vacuum.

Not true. The wave nature of matter is not in doubt, but vacuum fluctuations are absolutely nothing like the hard gamma rays from which we can make matter. For an analogy, think of a gamma photon as an oceanic swell wave. Vacuum fluctuations are then something like the tiny random ripplets on the surface of the ocean.    

The researchers simulated the frantic activity that goes on inside protons and neutrons. These particles provide almost all the mass of ordinary matter.

This is rather misleading. A proton is not some bag of "frantic activity".  

Each proton (or neutron) is made of three quarks – but the individual masses of these quarks only add up to about 1% of the proton’s mass. 

That's correct. See Gian Giudice's Zeptospace Odyssey. He's a CERN physicist:
 
Edit: note though that the frantic motion of quarks and gluons isn't quite right, because the gluons in ordinary hadrons are virtual. 

So what accounts for the rest of it?

E=mc².

Theory says it is created by the force that binds quarks together, called the strong nuclear force. 

It doesn't. The mass of a body is a measure of its energy content. See Einstein's E=mc² paper. 

In quantum terms, the strong force is carried by a field of virtual particles called gluons, randomly popping into existence and disappearing again. 

This is badly misleading. Virtual particles are not short-lived real particles that pop in and out of existence like magic. They "only exist in the mathematics of the model". See anna v's answer here. She's a retired particle physicist.    

The energy of these vacuum fluctuations has to be included in the total mass of the proton and neutron.

More misleading material. Virtual particles are not the same thing as vacuum fluctuations. In the oceanic analogy, a virtual particle is like a cubic metre of sea water. Did you see the last line?

So if the LHC confirms that the Higgs exists, it will mean all reality is virtual.

Reality is not virtual! I'm sorry but New Scientist is often sensationalist and misleading. I stopped reading it some years ago. 
A: Answer in relative terms: Yes they did. Relatively speaking, there is that which is "nothing", and there is that which is "something". In relativity, and to validate the concept of causality, there must be the existence of that which in one way or the other affects an observed event so that empirically measurable values of the event are detected as to change. Whatever change is detected, there must be an observed quantity of that which has the quality of causing that particular change. If anything at all happens, we must relate this happening to an actor of a particular action. The question in empirical investigation is then to establish a causal link between the observed effect and what caused it. The truth of a suggested causal relation is based on regularity and predictive value. So from a relative and scientific perspective, since there is obviously that which is something empirically real and observed as to be in constant change/flux, there must also be that which caused it to be so. Relativity can map cosmological evolution back to the electroweak epoch, but no further. The suggested "Grand Unification Epoch", coming before the electroweak, is still to be understood. Unification implies a situation where relativity is not all there is. There might also be a quality of absolute-ness and discrete-ness. That brings us to the complemetary answer.
Answer in absolute terms: No they did not. Absolutely speaking, there are no "fluctuations" in a moment of singularity. If it fluctuates, it is a single fluctuation. If not, we are assuming a scenarion of many initial states, not an initial state. Further, the concept of "nothing" as opposed to "something" is nonsensical least we agree upon what is required for the quality of detection. This is to ask; what is it with a hypothetical quantum/unit that makes it real in the empirical sense? Could there be a unit S which is true in the ontological sense of actually being S, but also being empirically unobservable, and in that sense "unreal"? Could there be that which only can be analysed based on an imaginary value, and that evolves in such a way as to translate the imaginary base value into an inceasingly real value?
This is important to contemplate, as is done in math's complex analysis, because in absolute terms; there is no possible relation of nothing vs. something for the simple reason of "nothing" being absolutely impossible. It defies reason and logic to assume there is the existence of that which is defined as non-existence. It is to say that existence includes non-existence, and we create conceptual uncertainty "out of nothing".
An alternative question could be:
Does the quality of fluctuating cause the effect of a quantity being of variable empirical value(s)?
To phrase the question thus is an effort to stress the importance of quality in the analysis of any quantity. It is to say quantum theory might lack a definite sense of quality. A quantum of reality is a statement of ontology. It is to say: there is that which is, and we can assign this fact a nominal value e.g. Unity, One, Singularity, initial state, God, i1,r0. The point is, a nominal value can be true while not being of any empirically observable, and thus analytical, value. As we all know, the imaginary part of complex analysis is related to rotation. Rotation/spin is not related to the quantum actor as such, but to its quality of action. It is a quantum of quality, and that quality is acting as rotation.
I suggest that the action of rotation can cause the effect of translating a quntum of this action as to project angular eccentricity perpendicular to bidirectional compression of its rotational axis. By this, it is not unreasonable to assume the quality of the quantum will generate/produce the emergence of empirically real values from av base value of a sphere/ball with imaginary axis equal to diameter i.e. symmetric. The nothingness of the base symmetry is then of empirical values, not ontological. With flattening of the quantum, empirical values of space/eccentricity will increase in magnitude all the way to full transformation at moment of 2D Surface. The discrete value, now an empirical value, might then be be opposite the initial and imaginary 1. Axis turned out,by rotation, to be horizon would be imaginary 0 = real 1.
Then we have turned eternity of zero space into a neverland of infinity.
Eternity is nowhere
Infinity is never
All of empirical values as being "real" are likely to be observed as continuously fluctuating between two extreme discretes, assumed here to be possible only in the special case of singular action.
Ergo: The fundamental and thus unconditioned force of rotation caused a singular and uniform compression and eccentricity of an initial quantum, equal to an initial quality. Thereby, energy as  kinetic and potential, is fundamentally equal and uniform, and in the singular case not the cause of change. The cause is force of rotation. The effect is that eccentricity is compressed as compression is eccentric. Further, empirical values to be observed at all requires a real observer which is, at the moment of observation, of values not the same as the values to be observed. This makes singularity unobservable, even if it is of empirical value, except for when being in the moment of perfect 3D symmetry. At that, it is void of any values but imaginary/axial "time". 
What also requires a real observer is gravitation. A singular body can implode vertically as it explodes horizontally, but cannot attract anybody but itself. When it does, it also rejects itself.
Thus, gravity is not a fundamental cause but a relative effect in a many body context.
To make scientific sense of the above, we must analyse a parity of such events. I suggest this dual space generates values of electromagnetism. Assumed to be liquid-like, the values of quanta will locate and relocate as it is forced to flow in vortex-like nodes acting as either "well" or "drain".
The lack of references is because I try to offer an alternative model/perspective, not to add complexity to the ones we already have. It is I who suggest there is only one fundamental force i.e. rotation, that there is no fundamental energy and why any initial state singularity must be known by having faith in our imagery. Abstractions in math and teology can  handle the singular action. Empirical science can not. Symmetry is of zero empirical value. 
My One Cent
