I've always thought that DLLs and shared objects are just different terms for the same thing - Windows calls them DLLs, while on UNIX systems they're shared objects, with the general term - dynamically linked library - covering both (even the function to open a .so on UNIX is called dlopen() after 'dynamic library').
They are indeed only linked at application startup, however your notion of verification against the header file is incorrect. The header file defines prototypes which are required in order to compile the code which uses the library, but at link time the linker looks inside the library itself to make sure the functions it needs are actually there. The linker has to find the function bodies somewhere at link time or it'll raise an error. It ALSO does that at runtime, because as you rightly point out the library itself might have changed since the program was compiled. This is why ABI stability is so important in platform libraries, as the ABI changing is what breaks existing programs compiled against older versions.
Static libraries are just bundles of object files straight out of the compiler, just like the ones that you are building yourself as part of your project's compilation, so they get pulled in and fed to the linker in exactly the same way, and unused bits are dropped in exactly the same way.