Well, for a start you probably mean compiled for an arch with 16 registers. It doesn't actually matter what arch the compiler ran on (assuming modern cross-compiler like GCC).

If a program uses 16 registers then it needs 16 registers. But note it's the program itself that decides to use a frame pointer, it's not being reserved by the operating system or something. Programs don't even have to use the stack pointer as a stack pointer, they could use all 16 as general purpose, but in practice almost all programs use a call stack (I guess all C programs must do, but you might be able to disable it if you make no function calls?)