Here is the home page for the GUFM model website. It also includes a link to a freely available pdf of the modern reference. Also of interest is the NOAA WEBSITE.
GUFM MODEL HOMEPAGE
NOAA PAGE
Note that this model is based upon catalogs of geomagnetic field measurements, these did exist prior to 1800 - although as you would expect their quality and quantity decreases the farther back in time you go. The model is trying to solve an Inverse Problem. A 'solution' represents the model that provides the best fit to the available observations. It therefore may be considered our best empirical knowledge of the historic geomagnetic field based upon our limited observations.
Results such as the figure that you have posted would be more useful if, in addition to the most-likely values, they also displayed an objective measure of uncertainty (such as confidence limits) in the figure.
To clarify these terms a bit - This study is based on historic records (mostly ship's logs). Empirical information about the pre-historic geomagnetic field over geological time-scales is available to the science of paleomagnetism through the application of our understanding of rock magnetism. Rock magnetism is the study of the magnetic properties of rocks, sediments and soils. The field of rock magnetism arose out of the need in paleomagnetism to understand how rocks record the Earth's magnetic field.
Actually, in a large solar flare particle energies can get up to 1 GeV, but the top energy of some particles is not really the issue. The issue is the flux of these high energy particles. A 10 MeV proton or electron pretty much rips through most spacecraft bodies, thus, their electronics are effectively exposed to particles at these energies.
The often associated coronal mass ejections (CMEs) produced in association with large solar flares carry with them enhanced fluxes of >MeV protons and electrons. These blobs of plasma and magnetic fields compress the Earth's magnetic field, which can induce DC currents in our power grids and expose geosynchronous (or GPS, I forget which orbit at the co-rotating altitude) spacecraft to the high levels of radiation. After the CME has passed, the effects are not over as they often induce a geomagnetic storm, which enhances the radiation belts and thus further exposes co-rotating spacecraft to high energy particles (thus the name "killer electrons" for the outer radiation belts).
I will add more later and include some links, but the point is that our magnetic field does a tremendous amount to prevent our lives from becoming incredibly complicated, as Timaeus eluded to.
Updated Version
Actually, in a large solar flare particle energies can get up to 1 GeV, but the top energy of some particles is not really the issue. The issue is the flux of these high energy particles. A 10 MeV proton or electron pretty much rips through most spacecraft bodies, thus, their electronics are effectively exposed to particles at these energies.
The often associated coronal mass ejections (CMEs) produced in association with large solar flares carry with them enhanced fluxes of >MeV protons and electrons. These blobs of plasma and magnetic fields compress the Earth's magnetic field, which can induce DC currents in our power grids and expose geosynchronous (or GPS, I forget which orbit at the co-rotating altitude) spacecraft to the high levels of radiation. After the CME has passed, the effects are not over as they often induce a geomagnetic storm, which enhances the radiation belts and thus further exposes co-rotating spacecraft to high energy particles (thus the name "killer electrons" for the outer radiation belts).
The Earth's magnetic field also helps protect our atmosphere from ionizing erosion. By that I mean that once an atom is ionized and exposed to the bulk flow of the solar wind, it will experience a conductive electric field ($\mathbf{E} = -\mathbf{V} \times \mathbf{B}$) and react like a pick-up ion. The force on the particle from such an electric field can easily exceed the gravitational force, thus freeing the particle from the atmosphere. Without the Earth's magnetic field, the ionized part of the upper atmosphere, called the ionosphere, would increase due to the addition of the solar wind's ionization effects. Currently, only charged particles with energies >10-100 MeV, neutral neutrons, or high energy photons (e.g., UV, X-rays, and/or $\gamma$-rays) are able to reach our atmosphere and contribute to the overall ionization.
It is doubtful that during a pole flip of the Earth's magnetic field that we would completely lose our atmosphere, considering several pole flips have happened in the past. However, the point is that our magnetic field does a tremendous amount to prevent our lives from becoming incredibly complicated, as Timaeus eluded to.
Best Answer
The aurora are emissions of light caused by the excitation of nitrogen and oxygen mostly by energetic (i.e., ~1 keV to few 10s of keV) electrons coming from the Earth's geomagnetic tail (i.e., anti-sunward direction). There are proton-driven aurora too, but they are fainter and more rarely observed (partly because it takes much stronger geomagnetic storms to produce enough proton precipitation into the atmosphere).
Note that electrons with this range of energies always exist in the solar wind, however the number fluxes are much lower than those that drive the aurora we all know and love. The solar wind plasma tends to have number densities of ~1-10 cm-3 and bulk flow speeds are typically ~300-500 km/s. To put that in perspective, that corresponds to a dynamic pressure of ~0.07-2.0 nPa, or more than 50 trillion times less pressure than one atmosphere at STP.
So if the Earth were to suddenly have no magnetic field, there would be a diffuse aurora seen (only by special instruments, as I explain below) at almost all latitudes but it would be extremely dim.
Part of the reason for the brilliance of the aurora is due to a sort of magnetic focusing during the precipitation process. So imagine taking a volume of space much larger than the Earth filled with electrons, accelerating them earthward, and then focusing them into an area that's a mere fraction of Earth's total surface area. The combination of the focused area and high fluxes results in a large contrast between the night sky and the aurora (which are actually much dimmer than a full moon).
No, not really. It would probably look more like what we see during very strong geomagnetic storms, namely, dim and diffuse light over much of the sky. However, I doubt anyone would see much as the solar wind is incident on Earth from the same direction as sun light (i.e., only daytime auroras on such an Earth).
Aurora on Mars
Mars has aurora like Earth, but it does no longer has an intrinsic magnetic field. There are surface remnant fields and it is about these that we see faint aurora. Yes, Mars does have an atmosphere, but its much more tenuous than Earth's. In fact, a neutral gas surrounding a planetary body is really all that is necessary to produce emission of light from impacting charged particles, i.e., this is basically to what the aurora amounts.
Aurora Elsewhere
We have also observed aurora on Jupiter, Saturn, Uranus, and I think Neptune. However, it is mostly in the UV light spectrum, not visible that we observe these.
Side Note
I should point out that while the loss of an intrinsic magnetic field would be very problematic, as I discussed at https://physics.stackexchange.com/a/214509/59023, it would not mean there would be no magnetic field.
There would be two types of ionizing radiation: electromagnetic radiation at UV or higher energies and energetic particles. The net result would be a very active ionosphere like that of Venus. This would create an ionopause and a bow shock similar to what Earth's magnetosphere does (the bow shock, not ionopause), but the bow shock would be much closer.
Let's assume Earth lost all of its atmosphere. Without an intrinsic magnetic field or atmosphere to ionize, the particles would mostly pass by the planet on balistic trajectories producing a wake similar to what is seen in supersonic wind tunnels. However, the solar wind does carry a magnetic field, so the particle trajectories around the blunt object would not be purely hydrodynamic (i.e., the Lorentz force still matters).