As Michael Brown mentioned in the comments, no one will explain this as well as Feynman (at least, no one we know of that's alive). But that doesn't mean your question doesn't deserve at least our attempt. So here is mine and I will try to keep this in the simplest terms I can.
(Aside: to all of the physicists reading this, I apologize in advance but in my simplification, I may intentionally omit or contradict the true physics. For instance, I doubt I'll be saying how the photon arises as a gauge boson in local U(1) symmetry).
- See (2)
See (1). Just kidding. I've lumped 1 and 2 together because to explain what a photon is is to essentially explain where it comes from. Hopefully, everyone reading this will be aware of the wave-particle duality that most(all) things enjoy. In that way, as mentioned, a photon is a particle in its own right. But does that mean that people can think of the photon as a tiny billiard ball? No, that would be silly. The photon is a wave packet that, for all intents and purposes, is indivisible. Consider a vibrating electron, it's motion one way or the other constitutes a current, which radiates a magnetic field. Since this magnetic field is changing as the electron speeds up and slows down, it induces an electric field that radiates outward. Since this electric field is also changing continuously, this in turn induces a magnetic field that radiates outward. Rinse and repeat. The result is a self-propagating combination of electric and magnetic fields travelling outward from the electron. This is EM radiation. The photon is the unit of an EM wave. What is one photon? Say we shine a laser, then we block half the beam with a metal plate. The other half still comes through. If there were only one photon in the beam, when you block half of it with the metal plate, none of it would come through. One photon is the largest amount of energy of an EM wave where this would still be true. In physics terms, we write it as $E=h\nu$. The energy of one photon of a wave is equal to h (a very small constant) times the wave's frequency. Because the photon is indivisible, we can say it represents the smallest unit of energy of that particular EM wave. A different wave would have a different smallest energy. To address what was mentioned in the comments, a photon may have full particle status, but it is not similar to an electron. Photons have no mass, they are not matter and, when you examine the properties of a photon, there is no denying that they are packets of energy in every respect; fluctuations in the background EM plane.
I'll admit, at first glance it does seem very silly for us to say electric and magnetic fields can produce action at a distance and then say, "no, you need photons to cover that distance and actually do the interacting". But it's true. At least, we can say it's true. Without teaching everyone about field theory and symmetries, my short answer to this would be that in advanced physics, we have a certain equation that originally didn't work out. As we've done a million times in the past, we had to modify this equation to work and we found that we could only do that by introducing a massless particle that mediates the EM force. Afterward, we noticed that this particle happened to have the same properties of a photon. In fact, if we tried the hypothetical situation where we assumed this particle was a photon, this one equation produced all of the laws and equations from electromagnetics that we already knew and loved. Thus, we said, "we're pretty sure this equation is the right one to use. We assumed this particle we invented was a photon, and it resulted in the equations and laws we have in the real universe. So this must be the way it actually is!". Having said that, we can never actually observe the photon as it mediates the force. This is simply because if we were to observe the photon, it would no longer be able to mediate the force because we have no method of observing a photon without destroying it.
I have already explained how they are created, how they are destroyed is much simpler. When a photon hits something, it can either reflect, transmit, or be absorbed. The latter is destroying it. When it is absorbed, this means whatever it struck (usually an electron) absorbs all of the energy of that photon. That's it.
How we know they exist... We know because we can do experiments with just one photon. We can see the effects of one photon. But most importantly, theoreticians say, "if a photon didn't actually exist, what would happen in some experiment? Well Jim, we would see outcome. And if they do exists, we should see different outcome." Then experimentalists perform the experiment and 10 times out of 10 we always see the outcome predicted by the existence of photons.
This particle/wave packet is called a "photon" because it was first theorized specifically about light. I believe "photo" is from the greek word meaning light and the extra n was added because all of the particles known at that date ended with an n (proton, electron, neutron. Why not photon?)
I'm willing to bet I've missed something important, so let me know. If I can merge what I've missed with the general form of the answer, I'll be happy to put it in.