C does have scoped variables, where a variable only exists within a pair of braces. When I was writing C code that got cross-compiled to 68000 assembler, I could see that the compiler just assigned space on the stack for that type of variable by moving the stack pointer down, and once the scope ended, the stack pointer was just adjusted back up.
That's quite typical for a C compiler, and C works that way on all common platforms (x86, ARM, AVR, ...).
C also has the static keyword, which says the variable is to be stored in the lexical scope, meaning it's created at the beginning of the run (perhaps on the stack, but not anywhere that's going to get re-used), and only initialized once, if that initialization happens during declaration.
No, static variables inside a function are NOT on the stack. They work like normal static variables, except that the name of the variable is known only inside the function scope. If static local variables were on the stack, their content would be lost on return from the function.
Example (on x86-64 Linux):
The printf output is quite obvious: the global and the static local variable follow each other without any gap, in the data area. The local variable is far away, on the stack, as is the function parameter.
Today I will gladly share my knowledge and experience, for there are no sweeter words than "I told you so". ;-)