Does the GPL have different requirements for statically vs dynamically linked modules with a covered work? (#GPLStaticVsDynamic)
No. Linking a GPL covered work statically or dynamically with other modules is making a combined work based on the GPL covered work. Thus, the terms and conditions of the GNU General Public License cover the whole combination. See also What legal issues come up if I use GPL-incompatible libraries with GPL software?
https://www.gnu.org/licenses/gpl-faq.ht ... uleLicense
If I add a module to a GPL-covered program, do I have to use the GPL as the license for my module? (#GPLModuleLicense)
The GPL says that the whole combined program has to be released under the GPL. So your module has to be available for use under the GPL.
But you can give additional permission for the use of your code. You can, if you wish, release your module under a license which is more lax than the GPL but compatible with the GPL. The license list page gives a partial list of GPL-compatible licenses.
https://www.gnu.org/licenses/gpl-faq.ht ... braryIsGPL
If a library is released under the GPL (not the LGPL), does that mean that any software which uses it has to be under the GPL or a GPL-compatible license? (#IfLibraryIsGPL)
Yes, because the program actually links to the library. As such, the terms of the GPL apply to the entire combination. The software modules that link with the library may be under various GPL compatible licenses, but the work as a whole must be licensed under the GPL. See also: What does it mean to say a license is “compatible with the GPL”?