Oh, wow. I'm sorry, but your roof is probably bad. If you can get the money you paid four years ago back (doubtful), I would, but you probably need to get that entire mess torn off all the way down to the decking, and probably quite a bit of the decking near the edges of the roof too. You can tell because the shingles look "lumpy" and have a rolling look to them -- like someone's melted them, especially near the edges -- and the valley is curved instead of sharp. I can also see the lines running across where different runs of roof sheathing panels meet (every 4' or so); that it's sagged in between these lines is a sign of sheathing damage.
Your roof isn't the shingles, so it doesn't matter that the shingles are only four years old. Your roof is actually the tar paper and ice and water shield that is underneath underneath the shingles. Multiple layers of shingles are bad; it means that they didn't replace the actual stuff that keeps the water out of your house, they just replaced the stuff that you see that goes over top.
If you go up into your attic when it's been raining quite a bit, see how much more of the underside of your roof sheathing is damp. If there's a lot, you need to get it torn off and replaced as soon as possible. If there isn't much, you have a few years to save up for it, but it looks bad now from the top. Make sure when they re-do it that they tear everything down to bare wood, remove all the nails and staples and junk, use Ice & Water Shield along the roof edges and in the valleys, and then put new 30 year tar paper and shingles over it. The tar paper and ice&water shield is what actually keeps the water out of your house. The shingles only protect the tar paper from abuse.
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.
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My shed has (better-sheltered) gaps. I believe my shed's gaps are on purpose, to enable air circulation. This prevents mold and mildew problems in the unheated space.
I am surprised that your shed has fiberglass insulation in the walls, but not in the ceiling. With no heat, and so many thermal bridges and air gaps, I would be surprised if the insulation noticeably affects the temperature inside the shed.