A few additions to Jeff's advice if you decide to replace the damaged drywall.
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.
Double check the thickness of your drywall if you are patching. If you replace a 5/8" with 1/2", you'll notice it.
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.
For starters, torch down is a rubber product. It should NOT be installed over asphalt shingles. Swelling and contracting will eventually allow the asphalt shingles to wear a hole in the torchdown membrane from the underside. The same applies for rubber membranes.
You are correct about the drip edge needing to be under the roofing material. This applies to ALL roof coverings.
Your roofer should have mentioned the soft spot prior to covering it up. That's complete irresponsibility. Roofers typically don't repair facia but it shouldn't be beyond a decent roofers skill set.
As for what the rafter you need repaired... Unless there's access from below, a portion of your new touchdown will need to be removed, along with the decking above. Unless the rafter is totally rotted out, I'd scab another 2x to the side of the rafter in question instead of removing it completely.
Additional details per OP questions:
The recommended overhang for torch down is 1/2".
For roof pitches below 12/4 torch down or rubber are my recommendation.
To my knowledge, it's not standard practice to double up material around the drip edge.
Since I'm unsure what your rafters are, I used a generic term "2x" ( pronounced 2 by ), which can mean anything from 2x2 to 2x12.
Yes, if there's any bad decking it should be replaced, but it can be spot fixed from rafter to rafter.
Additional Info
Decking is a generic term... Meaning your roof deck ( sheathing ). Spot fixing means cutting out the bad decking from rafter center to rafter center.
I've never heard of Timbor. I ALWAYS replace rotten/dry rotted decking.
To repair the damaged rafter, screw or nail a new rafter to the side of the bad one.
Best Answer
Laying down a roof starts from the bottom and goes toward the peak. Trying to repair something like the drip edge or lowest run of shingles is almost never going to give a good result without starting over.
If you need to replace the fascia board, you mentioned rotting, usually also involves stripping the roof, especially when there is an existing metal edging installed over it and maybe even nailed into it.
In my experience in working on roofs hanging the shingles a small amount over the drip edge does not significantly increase the ability of wind to rip the shingles off the roof. Of course it is best if the roof is laid down with the very first course using a starter roll instead of shingles. Good quality starter roll has a row of tar strips a short distance up from its lower edge to which the tabs of the first row of shingles are meant to bond.