extern "C" doesn't really change the way that the compiler reads the code. If your code is in a .c file, it will be compiled as C, if it is in a .cpp file, it will be compiled as C++ (unless you do something strange to your configuration).
What extern "C" does is affect linkage. C++ functions, when compiled, have their names mangled -- this is what makes overloading possible. The function name gets modified based on the types and number of parameters, so that two functions with the same name will have different symbol names.
Code inside an extern "C" is still C++ code. There are limitations on what you can do in an extern "C" block, but they're all about linkage. You can't define any new symbols that can't be built with C linkage. That means no classes or templates, for example.
extern "C" blocks nest nicely. There's also extern "C++" if you find yourself hopelessly trapped inside of extern "C" regions, but it isn't such a good idea from a cleanliness perspective.
Now, specifically regarding your numbered questions:
Regarding #1: __cplusplus will stay defined inside of extern "C" blocks. This doesn't matter, though, since the blocks should nest neatly.
Regarding #2: __cplusplus will be defined for any compilation unit that is being run through the C++ compiler. Generally, that means .cpp files and any files being included by that .cpp file. The same .h (or .hh or .hpp or what-have-you) could be interpreted as C or C++ at different times, if different compilation units include them. If you want the prototypes in the .h file to refer to C symbol names, then they must have extern "C" when being interpreted as C++, and they should not have extern "C" when being interpreted as C -- hence the #ifdef __cplusplus checking.
To answer your question #3: functions without prototypes will have C++ linkage if they are in .cpp files and not inside of an extern "C" block. This is fine, though, because if it has no prototype, it can only be called by other functions in the same file, and then you don't generally care what the linkage looks like, because you aren't planning on having that function be called by anything outside the same compilation unit anyway.
For #4, you've got it exactly. If you are including a header for code that has C linkage (such as code that was compiled by a C compiler), then you must extern "C" the header -- that way you will be able to link with the library. (Otherwise, your linker would be looking for functions with names like _Z1hic when you were looking for void h(int, char)
5: This sort of mixing is a common reason to use extern "C", and I don't see anything wrong with doing it this way -- just make sure you understand what you are doing.
Best Answer
So here's a fun little detail about Eclipse we didn't know - specific source files themselves can have their own settings.
If you look at your source files and see the following wrench icon, this means that file has some settings which may be contrary to your project-specific settings.
As a result, our source-specific settings were overriding the include paths, so for those files, none of the headers could be found. The best part is that Eclipse doesn't tell you in any way more explicit than this - the wrench icon does not offer a tool-tip to explain what's going on.
Your header files will be found now, provided that you've written the correct include paths in your project settings.
For reference, we're using Eclipse Kepler - I'm not sure if they fixed this and added more info for users in newer versions.