You will want to contact the manufacturer of the product to get their recommendations on the type of fasteners, the fastener spacings, and the necessary penetration of the fasteners.
Yes, you will need flashing around the entire perimeter if you wish the roof to be waterproof.
Any moisture issues related to the standing water incident would have manifested itself long before now. Whatever the current issue is, it's unrelated to the previous issue. In that time, it is possible a new leak developed. It's also possible the dark areas are related to interior humidity and unrelated to the roof. In cold weather, the ceiling surface under rafters is measurably cooler than the nearby portion under insulation alone. This can cause condensation only under the rafters. As you know, condensation can lead to mold.
Can you tell if the darkening is due to actual mold? The damp surface can attract dirt that darkens the area even though no mold is growing. Even though the dirt can stain, attempting to clean an area with plain soap and water would yield a very different result than if mold is actively growing.
If mold is actively growing, it is time to consult with a mold professional. They will determine the cause, mitigate the mold, and suggest how to prevent it from happening again. If you want to investigate further before going that route, the only reliable way to determine if the rafters are getting wet is direct inspection.
Cut a small inspection hole where you think it's the worst. If it's wet up there, a lot more ceiling is coming off, so don't bother patching the hole. The roof is leaking. It needs to be repaired and all wet drywall and insulation needs to be replaced. If it's all dry, breathe a sigh of relief. The need to patch the hole is much less than it could have been. Be sure to adequately seal the damaged vapor barrier.
If the rafters are dry, the only thing you can do to reduce condensation on the ceiling is reduce the overall humidity of the house. First try selective ventilation and lifestyle adjustments. Consider bumping the thermostat up a bit. If that's not enough, look into continuous ventilation through an air to air heat exchanger.
A far as replacing stringers, that cannot be done with comp roofing unless a full cold roof is installed. It is normal practice for concealed structural spaces filled with insulation to not be ventilated. It sounds like the roofers followed conventional building practice.
Best Answer
Its odd looking, but the batten support lets the tiles 'notch' into place. You'll see the same support for clay tiles, also.
Because of the 'hanging' aspect of the tile, most locales only allow tile roofs for pitches between 4/12 and 6/12 (rise over run). Lower pitches are possible, but then waterproof membranes are required. In those cases, sheathing support for the membrane would be needed.
You don't mention it, but not using sheathing is usually only done in warmer climates, where summer heat is the main problem.