Confusion regarding photons? I'm still in high school, and while I can't complain about the quality of my teachers (all of them have done at least a bachelor, some a masters) I usually am cautious to believe what they say straight away. Since I'm interested quite a bit in physics, I know more about it than other subjects and I spot things I disagree with more often, and this is the most recent thing:
While discussing photons, my teacher made a couple of statements which might be true but sound foreign to me:


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*He said that under certain conditions, photons have mass. I didn't think this was true at all. I think he said this to avoid confusion regarding $E=mc^2$, however, in my opinion it only adds to the confusion since objects with mass can't travel with the speed of light, and light does have a tendency to travel with the speed of light.. I myself understand how photons can have a momentum while having no mass because I lurk this site, but my classmates don't.

*He said photons don't actually exist, but are handy to envision. This dazzled my mind. Even more so since he followed this statement by explaining the photo-electric effect, which to me seems like a proof of the existence of photons as the quantum of light. He might have done this to avoid confusion regarding the wave-particle duality.
This all seems very odd to me and I hope some of you can clarify. 
 A: He is doing a good job trying to communicate the weirdness of Photons, but a poor one at being consistant. It's difficult to communicate how strange they are, without using the relationships you described.
First of all photons have no mass. At rest. The concept of describing a photon at rest is a bit weird even as there is no such thing. 
Once your talking about photons in motion it is tempting to say they have mass, but most physicists these days don't speak of it this way. Rather its best to say it has ENERGY. While people like to say things can have relativistic mass, this is incorrect, mass is a constant: the mass a thing has never changes no matter how fast you travel: rather the energy to accelerate it increases (inertia).
So a photon has no mass: but it does have momentum. This is the fascinating thing about light. And leads to the fact that momentum is not always related to mass as you mentioned.
Examine $E=mc^2$ in more detail and the correct equation actually is $E^2 = m^2 c^4 + p^2 c^2$.
As photons have momentum related to its energy/frequency this is fine.
The best way to think of light is as photons. Light is a photon, that is the fact, BUT light waves are only a model. The appearance of wave phenomenon is because quantum mechanically light interacts probabilistically, and this nature allows it to display wave like properties. 
The science of Quantum Electro-Dynamics examines this strange behavior.
For now think of light as a photon and a wave, it allows every day behavior to be modeled well and is perfectly fine.
But in reality light is a photon particle (this is why the photo-electric effect works) that when described using Schrodinger's Equations (a description of probabilites) can be transformed into a description of a wave of Electro-Magnetism. 
A: Regarding your two points:

He said that under certain conditions, photons have mass.

Massless particles move at the speed of light $c$ in vacuum.  By his statement your teacher may have been alluding to the fact that photons travel at speeds slower than $c$ when they travel through media, like glass for example.  However, I would rather phrase this as something like "in the transmission of a photon through a medium, the photon's transit time through that medium is such that it travels as if it had a mass."  The travel of photons through media is a rather complex affair, which I don't fully understand, involving interactions with charged particles in the medium and quasiparticle states, and I'm not sure to what extent the incident photon even retains its identity whilst in the medium (maybe that's another question).

He said photons don't actually exist, but are handy to envision.

There is no question that photons exist.  Particle physicists deal with "hard" (i.e high energy) photons, which behave like particles - scatter off other charged particles etc.  Good evidence that "soft" (low energy) photons exist comes from antibunching experiments.
A: 1) I would refer your high school teacher to some of the good answers found in this physics stack exchange question. The answer with the highest votes is really good and is referring to the electroweak theory which governs the electroweak section of the Standard Model.  
General Relativity respects special relativity, and therefore respects that the invariant mass of the photon is zero, since the photon is described by a null-like vector in its rest frame.
In particle physics, as discussed in the links above, the current understanding is that mass is a measure of the relative interaction strength of a particle with the Higgs field as mediated by the Higgs boson.  The photon does not directly interact with the Higgs boson, and therefore has no mass. 
2) As far as visualizing the photon, I would venture the easiest way is to think in terms of classical EM theory (which is a gauge theory btw) where we consider the orthogonal oscillating electric and magnetic fields as representing the photon.  
A: *

*Photons are massless.  This should not cause confusion with $E=mc^2$ because the expression for the relativistic energy of the photon is $E = h \nu$, there $\nu$ is the photon's frequency and $h$ is Planck's constant.  You can also understand the relativistic energy of the photon by noting that $E = pc$ where $p=\hbar k$ is the magnitude of its momentum, and photons possess momentum, as you point out. $k$ is the photon's wavevector and $\hbar = h/2\pi$ is the reduced Plank's constant. To tie this all together, we have the formula $E=\sqrt{m^{2}c^{4}+p^{2}c^{2}}$. 

*Photons exist!  As you point out, the existence of photons has physical consequences that can be measured.  Perhaps, as you also mention, your teacher is trying to insist that you resist thinking of photons purely as particles (which is probably not such a bad idea), but the statement "photons don't exist," in my opinion, should be considered just as fallacious as "the keyboard on which I'm typing doesn't exist.  
You're being prudent by taking everything your teacher says with a large grain of salt.  In my experience in physics (heck in life as a whole), it's good to take everything that anyone ever says with a grain of salt (including my response for that matter).
Cheers!
A: $E = mc^2$ is a popular formula but only valid in a special case when "the total additive momentum is zero for the system under consideration" $^1$
The more general "energy-momentum relation" is:
$E^2 = (mc^2)^2 + (pc)^2$
(Also, here's a  neat little video for your classmates ;) ->
http://www.youtube.com/watch?v=NnMIhxWRGNw )
You can read more about this and the correct four-vector notation under:
$^1$ http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence
Even today in the time of quantum optics, there are still quite a lot of papers, that try
to grasp what a "photon" is, e.g.:


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*What is a photon - David Finkelstein

*Light reconsidered - Arthur Zajonc

*The concept of the photon - revisited - Ashok Muthukrishan, Marlan O. Scully, M.Suhail Zubairy


and many more all together in this nice review:
http://www.sheffield.ac.uk/polopoly_fs/1.14183!/file/photon.pdf
Or in the words of Roy Glauber:
"A photon is what a photodetector detects"
