How GCC handles built-in function

I have trouble understanding GCC built-in functions, and feel very confused.

• What is the difference between a library function and an built-in function?

• Is there something a built-in function can do but a library function cannot?

• Can I write a library function that performs the same task as the built-in function printf? How can I tell the type of the input parameters (%f, float or double)?

• Machine instructions of GCC built-in functions are not stored in a library, right? Where are they?

• When doing linking, how can you control where to put these built-in function codes?

• Why sometimes I can error messages like "undefined reference to __builtin_stdarg_start" when doing linking

  // main.c
  #include <stdio.h>
  int main(void) {
    printf("hello world!\n");
    return 0;
  }
  

  gcc -c main.c, nm shows that there is no symbol printf in main.o, (only main(T) and puts(U)) , why?