Let's take these separately
Are all electrons always in pairs except the final single one if odd number electrons are considered?
For a ground state atom, then this depends on the sort of shell you're looking at. If you have an odd number of electrons, the simplest sort of ground state will look like this.
In this case, if you have unpaired electrons other than the final one, like, say,
then you have essentially promoted a core electron to a valence or higher shell, and the atom is in an excited state. This is perfectly all right but the state is unstable and it will eventually decay radiatively to the ground state.
Things get more complicated, however, if you have half-full shells at the same energy. In this case, you can have multiple unpaired electrons in different orbitals of the same shell, as long as those orbitals are degenerate in energy.
The apparatus to decide which configurations will have lower energies are known as Hund's rules. In general, you can and do get unpaired electrons in both even- and odd-$Z$ atoms. For examples, have a wander here.
Is there a binding force due to the opposite spin and magnetic moments of this electron pair? If so what is its magnitude and is it significant compared to electric field effects?
If electrons are in different shells, then the differences in their interaction with the core will dominate, and you get the first situation I described. If they are sitting inside a degenerate shell, then there are multiple interactions between them which compete with each other:
- The electrostatic interaction between them depends on how far apart they are. In general, if a configuration is spatially antisymmetric in exchange then the electrons will be further apart than in symmetric configurations.
- The symmetry of the wavefunction also affects how effectively the electrons can screen the nuclear charge for each other.
- Each electron also has a spin-orbit coupling, which depends on both its orbital and its spin angular momentum. This is obviously affected by how one apportions spin and orbital angular momentum in a configuration.
- There are also magnetic interactions between the electrons, which depend both on the relative directions of the spins and on their average relative distance.
In general, the running order is electrostatic > spin-orbit > spin-spin effects. The detailed calculation of what dominates is more involved but this is well described by Hund's rules. The Wikipedia article has a good explanation of the hierarchy of effects.
Can individual electrons from a pair be excited to a higher energy level or are they both excited together?
In general, a double excitation is much harder to do than a single excitation. Double excitations are generally unstable: it often takes more energy to excite a second electron than it does to remove the first excited electron; such states are known as auto-ionizing states. (In fact, even single excitations can do this: exciting a core electron is generally harder than ionizing a valence electron, so a core-excited atom will generally auto-ionize. This is known as the Auger effect.)
Is it easier to excite a single solitary electron from an orbital than one from a pair?
This is sort of tricky because it depends on which sort of pair you're considering. If you've got something easy like
then in general it will be easier to take the single electron simply because all the paired ones are in lower orbitals. Things are again trickier in half-full shells, so if you're trying to determine, for example, whether it's the red or the blue electron in something like
that will be ionized, then you need to do a careful calculation of the energies of all the possible final states to see which one will be easier to get to. With enough photon energy, however, you can promote any of the electrons in the diagram anywhere. (On the other hand, some transitions might be dipole-forbidden so they'll be much harder, or might require multiple photons. Still, you can pretty much make any eigenstate you want.)
Does the spin of the exciting photon impact which electron is excited?
Yes, but in general it depends on the situation. It's hard to say without having a more specific situation. The rules for what can and cannot happen with a given type of radiative transition are known as selection rules; in general they depend on the spin, orbital, and total angular momentum of the initial and final states, and on the multipolar order of the transition. In addition, two- or more-photon transitions are possible, further complicating the picture.