Help identify these protrusions on the leaky roof

A few additions to Jeff's advice if you decide to replace the damaged drywall.

1. I'd suggest you look into waterproof drywall or mold resistant drywall. There are various products that do a better job than the typical green board that you would use in the bathroom that will help if you are installing in an area that has a lot of exposure to the elements.

2. Double check the thickness of your drywall if you are patching. If you replace a 5/8" with 1/2", you'll notice it.

3. Use screws, especially on the ceiling. They take more time, but hold better and are less likely to cause the nail pops that you are seeing.

That being said, I think it's entirely possible that you're just more aware of drywall issues and there may not be anything wrong. To know for sure, I'd go up to the attic, pull up the insulation where you suspect damage, and look for water rings or other mold growth. Use a flashlight, no need to feel with your hands (and risk falling through). While you're there, check the insulation for blackening that would indicate mold is growing there, and if so, replace any damaged pieces.

Your ability to see the joints between the drywall is just a sign of not enough paint and/or primer. And popped nail heads are normal as a home moves. My personal plan is to replace all the popped nails with screws and double up on the spacing the next time I paint to minimize this issue.

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.