Here's an answer from a non-particle physicist to complement what (former) professional particle physicist Anna V has written.
"Real particles" enter and leave Feynman diagrams. Therefore, in principle, they can be detected in an experiment - they are the "terminals" of a Feynman diagram: ports through which we can "see" the system within.
In contrast, the path of a virtual particle begins and ends within a Feynman diagram. It has no "free ends" dangling over the "boundaries" of the diagram and is therefore not directly measurable. We can't detect them in experiment.
None of this is likely new to you. You're still left wondering what reality we can ascribe to virtual particles, if we can't directly detect them. You can think of virtual particles more literally as Feynman liked to do, or you might try this approach: I personally like to think of them a little more abstractly as simply as mathematical terms in a perturbation series.
A good starting point to visualise this gist is the kinds of ideas explored in the following papers:
as well as the works of the late Hilary Booth of the Australian National University.
This is not standard QED and it is very specialised and contrived: think of it as an illustrative "Baby QED" for someone (like me) who hasn't mastered quantum field theory. We consider here the system of one electron, a proton (the latter thought of as a classical particle, simply setting up an inverse square electrostatic field in a Hydrogen atom and the "virtual photons" that are swapped between them. The electron in the classical potential is of course simply described by the first quantised Dirac equation. Now we add the electromagnetic field by adding Maxwell's equations and coupling the system as follows:
$$\gamma^\mu\left(i \partial_\mu - q A_\mu\right) \psi + V \psi - \psi = 0$$
$$\partial_\nu F^{\nu\,\mu} = q\,\bar{\psi} \gamma^\mu \psi$$
$$F_{\mu\,\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$
with the Lorenz Gauge
$$\partial_\mu A^\mu = 0$$.
The first equation is the Dirac equation, the second Maxwell's equations with a charge / current (4-current) distribution determined by the probability density of the Dirac electron. The third relates the Maxwell tensor (containing the $\vec{E}$ and $\vec{B}$ fields) to the four-potential, which couples back into the Dirac equation through the "gauge covariant derivative". So we have a rather elegant, but thorny to solve, coupled nonliear system.
In the papers, the equations lead to a fixed point problem $X=F(X)$ of a certain integro-differential operator $F$, which is contractive, so the solution is the limit of the sequence:
$$X_0,\, F(X_0),\, F^2(X_0),\,\cdots$$
and can thus be solved nonperturbatively, by the contraction mapping principle and it gives an infinite series of terms corresponding to virtual pairs too. It yields an exact solution which is an infinite series, what a mathematician would call the Peano-Baker series (see Baake and Schlaegel, "The Peano Baker Series" and it is what a theoretical particle physicist would call (I believe) the Dyson series.
Now the terms in this infinite series are $X_0$: Dirac's solution of the Hydrogen atom and the higher order terms are iterated integral operators: these iterations can be thought of as the perterbations wrought by one "virtual photon", the next term involves virtual photons and virtual pair production followed by virtual pair annihilation and so forth.
The "Virtual particles" in this viewpoint can be thought of simply as an evocative "mnemonic" to the structure of the mathematical terms in the infinite series.
There is only one kind of photon.
Indeed, when we describe elementary interactions between two electrons for example, we call the photon "virtual" as opposed to a physical photon that might exist outside of this process.
Still, these are the same particles, i.e. excitations of the same fundamental field, as the photons that make up light for example.
Again, virtual photons can only appear in the context of a direct interaction between charged particles, while real photons are the electromagnetic waves send out e.g. by excited atoms. Macroscopic (constant) electric and magnetic fields are coherent states of virtual photons.
Regarding the electroweak unification you seem to have a misconception. In the unified theory there is no electromagnetism any more, but only the electroweak force, which has four force carriers: The $W^\pm, W^0$ and $B$.
The Higgs field couples to all of those, giving mass to the $W^\pm$ and to a linear combination of $W^0$ and $B$, which we call $Z = \cos\left(\theta_W\right) W^0 + \sin\left(\theta_W\right) B$, while the orthogonal linear combination $\gamma = -\sin\left(\theta_W\right)W^0 + \cos(\theta_W)B$ remains massless.
So the photon is defined as the boson that remains massless after electroweak symmetry breaking.
Best Answer
You have to realize that when we are speaking of photons, we are speaking of elementary particles and their interactions are dominated by quantum mechanics, not classical mechanics, and in addition special relativity is necessary to calculate anything about them.
In general, we know about elementary particles because we observe their traces in detectors for almost a hundred years. We never see an electron, or a proton in the way we see a particle of dust.
This is the most visual detector, a bubble chamber photo of electromagnetic events.
Now lets see about your questions:
That is not the way we arrived at this conclusion. A very large number of controlled scatterings, which is what this picture shows, of electrons on matter have been studied over the years and the theoretical framework of calculating the probability of the scatter and the angular distributions has been very well developed for years. This involves mathematics which cannot be handwaved. To start with, the crossection of an electron scattering on an electron can be written in a series of convoluted integrals which can be pictorially represented by Feynman diagrams. In those Feynman diagrams, the propagators of the interaction between the incoming and outgoing particles can be thought as virtual photons because they carry the quantum numbers of the photon but are off mass shell. So it is a convenient mathematical identification which defines virtual photons.
Anything between the incoming vertices and the outgoing vertices is virtual, and their reality depends on the correct representation of the quantum numbers for the exchanged particle, in this case photon quantum numbers.
Virtual photons are not like balls, they are off mass shell, they are useful a mathematical construct .There is an interesting analog though where two boats throwing balls at each other represent the repulsive forces, and boomerangs the attractive.
There exist cosmic rays of all energies, the cosmic accelerator, and elementary particles were first seen in emulsions exposed to cosmic rays in high altitudes, for example the pion was thus discovered. So any process seen in accelerators can be found if looking hard enough in cosmic rays. Accelerators allow detail and exact measurements of crossections and branching ratios etc. because of the high statistics possible.