Wood – What could the discolored areas on the underside of the roof be

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.

There could be several factors that may be contributing to the situation. It sounds like an air quality problem. This could be caused by airborne mold spores, chemical contamination from bad paint or flooring adhesives (VOC's) etc. , CO from a malfunctioning heating or A/C unit or something from outside getting in. I have had to address this problem with some customers in the past. We contracted an air quality testing from a local environmental testing lab. In your case, I would first consult with the management of the property. Explain the situation, confirm your conversations in writing, (document, document, document) and see if they will contract this service. If they won't for some reason, then I would have it done privately. Depending on the results and plan for mitigation, these costs should be reimbursed by the landlord. At minimum, it would give just cause to break the lease. If the results are bad and the management won't work with you, share the results with your local code enforcement office as they may help you and require the management to fix the problem. Once the type of irritant is identified, then the cause will be much easier to find. Good Luck.