If you don't have a problem with water pooling any more, the real answer is probably to have a professional roofer look closely at the roof both on the inside and outside. The problem with a lot of roof leaks is that where the water shows up in the house isn't where it comes in. Water has an annoying tendency to run along internal beams and such, then come into the house far away. You can go over any edges with new tar and hope you catch it but in my experience this approach ends up being like playing whack-a-mole, you patch and wait for it to rain, then patch more and wait more, etc.
So I would say if you can do it yourself, look closely at the underside of the roof as well as the outside. If you don't find anything, a professional roofer is probably in order.
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.
Best Answer
Oh dear. That's an unfortunate detail. :( I'm afraid DA01 is actually correct, the proper solution is tear out and properly rebuild. The proper approach to this would have been to either stop the barge end (the gable end overhang) several inches above the lower roof so the bottom could be properly enclosed and the lower roof properly flashed. This would not be a good detail architecturally though.
Another approach would be to fully extend the barge end so that it is actually fully framed into the lower roof, then flashing applied around the intersection. The barge end trim could extend a bit over the flashing, stopping short of the lower roof an inch or so. But the main structure is actually physically connected to the lower roof.
Those are the correct options, which involves a lot of remedial work and undoing much of what has been done in this area. If need be, you could do a stop gap treatment that will seal off the gap for now. This is far from ideal, and proper treatment should certainly be done when it comes time to re-roof.
This would involve fitting carefully shaped wood shims into the gaps to give the appearance the barge end had been framed into the lower roof. The outward facing shims could be toenailed in place, but the parts under the soffit would be just a pressure fit or sort of glued in with caulking. There will likely still be small gaps that need to be filled with caulk. Use a paintable caulk so when it's done and painted, it will not appear to be a patch job from any distance.
I don;t like this final option at all, but I understand if you are unwilling to do a proper fix at this point and thus offer this suggestion.