It's been a while since I used the syntax in anger, but basically....
"inline" is a compiler directive which says, do not vtable or pointer reference the function, do NOT use 'call', instead copy the resulting assembler for the function into "this" function instead. It saves you a TON of processor time. The function call process can be quite lengthy, especially if you are using C++ templates/abstraction/inheritance. Lots of stack push/pop'ing and potentially a page fault or two, resulting in context switches back to the kernel, and some MMU page table reloads etc. etc. etc. Basically a function call can add 100s or 1000s of instructions into the process. That can be avoided if you just "inline" the assembler verbatim. There are obviously caveats and it's use needs clear understanding to avoid unforeseen "scope" related issues and memory mamangement / register use etc.
As I said I haven't used the inline keyword that much lately, but I think you question around "static" and "extern" and more related to forward declarations of function pointers.
To give you an example of where I have seen it used, why and how.
Stock exchange. We received trade messages translated them, checked a few "fat finger checks" and wrote them out "on the wire". A HUGE amount of the critical path was taken up splitting the string into key=>value pairs on a delimiter.
The std lib functions for splitting the string ended up taking up about 50% of the processing time wire to wire. First we inlined some of the functions, removing the stack churn, which sped it up.
Then someone found code in another project which was using an ASM file to do the splitting of the string and leaving behind a lovely 2 dimensional array of key->value pairs.
The ASM went directly to the memory for the string and used CPU extensions to iterate of it moving pointers into the 2D array as it went. Using the CPU's built in iteration functionality gave huge performance gains and the CPU can pipeline, predictive and parallel execute the microcode.
We "extern"'ed the asm entry label. Declared it as an "extern "C" inline" function. The only complicated part, if I recall correctly was passing it the base pointers for the string and output tables, there is some really ugly looking syntax (in GCC) to populate registers before handing over to asm.
The linker takes care of the "extern", but I'm honestly not sure if it's the compiler or the linker that does the static code copy. I suppose that comes down to what executable type you are producing, like ELF of DLL for example.
