In case you haven't noticed, weather varies ;) This alone could account for the different effects each winter. But I think you're right, other factors are at play here.
Ventilation. Your gross vent area is 1/100 the attic area, accounting for louvers, let's say about 1/150th of attic area. If you do not consider the gable ends as cross ventilated, it is barely adequate. But as a cross ventilated arrangement, it is more than adequate. Will more help? Yes. Will it be effective? Harder to say. More on this later.
Moisture. Frost on nails alone is not necessarily cause for alarm. Black mold growing where the ventilation is less effective is definitely cause for concern. I'm no mold expert, but you should at least spray some moldicide on it while you devise a more permanent solution. Since the attic is always warmer than outside air, for there to be enough moisture to condense in the attic, the moisture had to come from either inside the house or from a roof or plumbing leak. Running a humidifier frequently would significantly contribute to the amount of moisture migrating into the attic. You don't mention anything about a vapor barrier, you quite likely do not know.The amount of moisture you observe tells me you don't have an effective moisture or vapor barrier. Not only can this cause mold, but it will also diminish the effectiveness of the insulation.
Why frost on only one side? Most likely solar access. The nails are colder on one side because the roof angles are different. This has significant effect on roof surface temperature, even under a good layer of snow.
Not only can vapor migrate through the ceiling, it can be carried in through air leaking around openings in the ceiling, in particular the pull down steps. If the perimeter is not weatherstripped, much moisture (and heat!) can escape this way.
What to do. Your primary focus should be to limit moisture migration into the attic. This will not only solve the condensation problem, but also limit heat loss. Seal all ceiling penetrations. Not only the steps, but plumbing and electrical penetrations as well. Apply an effective vapor barrier. This is not easy. You don't mention your ceiling finishes, this can make it easy or difficult or expensive. You could simply sandwich plastic between the current ceiling and a thin new ceiling finish. If you have an expensive existing finish, you may need to pull up the attic insulation and spray a barrier down on the back surface.
Note that if your household generates a lot of moisture, a good vapor barrier could cause you to have problems with condensation in your ceiling finish. The only solution to this is reduce moisture generation and increase whole house ventilation, ideally through a heat exchanger of some sort.
Once you've reduced the moisture migration, you may not have a mold problem in the attic any more. You may not need any more ventilation at all. The areas near the eaves obviously are not well ventilated, so the addition of soffit vents is the best approach if more is needed. You will need to drill many holes in the separation board to gain free communication with added vents.
If that works out to be too much effort, adding ventilation elsewhere can still benefit the lower areas, though not as well as soffit vents. Anything that increases ventilation will help. More static vents, power vents, ridge vents, turbine vents, they all have their own pros and cons, but they will all increase ventilation. You can do just one measure or a combination of measures. There is no right solution, but there is something that will make more sense than the other options.
Many building codes require at least 1/300 of the attic area in free openings for ventilation when cross ventilated, which it sounds like you have. If not cross ventilated, you need 1/150 of the area. Be sure your soffit vents are not obstructed by insulation. If the ridge and soffit vents meet this requirement without the gable ends, you might consider closing off the gable ends.
My guess is they were put there for a reason and you will not have adequate free area without them. I don't think you get extra credit for turbine vents though they can move more air because they don't move much air in still conditions. You should avoid mechanical ventilation if at all possible IMO.
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The reason eaves have this open space is so that moisture doesn't build up in the attic and cause rot. Completely sealing them off is generally not a great idea.
Attic insulation generally goes on top of the ceiling surface -- the space between there and the roof will generally be outside of the temperature-controlled part of the house anyway. It shouldn't really matter if the attic itself gets too hot or too cold. The main issues are vermin and moisture. There are vents you can install, and wire mesh will help with the vermin.
If you want to insulate a larger space than foam can handle, there are things called foam backing rods you can buy to fill large gaps -- once you've inserted it, there will be a much smaller gap between the foam backing rod and the surfaces, which can be filled with spray-in foam.