Mild water damage on drywall from leaky roof
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.