What is the cross-section size of a photon?

How "wide" is a photon,

The photon is a point particle in the standard model of particle physics. It has no extent to be described with "wide". Interactions of photons with charges or magnets can have a "width" in the sense of "measurable range"

if any, of its electromagnetic fields? Is there any physical length measurement of these two orthogonal fields, E and M, from the axis of travel ?

The photon has no electromagnetic fields . It has energy equal to $h\nu$ where $\nu$ is the frequency of the classical electromagnetic radiation, and $h$ Plancks constant. Its relations with classical electromagnetism come through its wavefunction, as it is a quantum mechanical entity. The complex conjugate squared of the photon's wavefunction gives the probability of finding the photon at $(x,y,z,t)$.

The classical wave can be shown mathematically to emerge from the superposition of very many photons of the given frequency $\nu$.

When a photon hits a surface, and is absorbed by an electron orbital,

It is absorbed by an atom (or molecule or lattice) by giving the energy (within the specific line width) to change the orbital of the electron to a higher energy level

this width comes into play,

No, it has nothing to do with the case as there is no such width identified with the photon.

as there could have been more than one electron that could have absorbed the photon?

No, the energy levels are what exist in the atom, and they are uniquely identified with quantum numbers also. The only width would be the size of the atom (molecule, lattice) and the quantum mechanical probability calculated with the given boundary conditions.

Depending on how you define size, a photon can be considered either pointlike or an extended object. It's pointlike in the sense that, unlike a proton, for example, it has no internal structure (as far as we know) that would give it a basic size.

On the other hand, a photon can exist in the form of a wave-train that's arbitrarily large. (There are some niceties involved in worrying about how to say this in a technically correct way, since QFT says that there is no position-space wavefunction for the photon, but I don't think that's particularly relevant here.) For example, the light from a typical pen-pointer laser has coherence lengths on the order of millimeters or centimeters. This is, in some sense, the "size" of the photon. There is no upper limit on the size of the photon. One way to see this is that the only scale in quantum mechanics is provided by Planck's constant, which can't be converted to a length scale because it doesn't have units of length.

When a photon hits a surface, and is absorbed by an electron orbital, this width comes into play, as there could have been more than one electron that could have absorbed the photon?

In the Copenhagen interpretation, we describe this by saying that the photon's position was measured, and therefore the wavefunction collapsed. (Again, this is modulo the facts referred to above about QFT.)

For a back of the envelope calculation, it is often useful to associate with a particle it's Compton Wavelength. This is generally the most accurately you can know an object's position due to the uncertainty principle.

Other related length scales are usually more useful, factoring in features of the particular interaction in question. This can be expressed as a Mean Free Path, basically the flight time between interactions or a Scattering Cross-Section, measure of the rate of interactions.