1st: Sure. You don't really need an operating system just to burn some cycles.
You might need some kind of OS support if you want to load or store files or data,
manage input or output, but this can also be done calling BIOS functions directly:
read key from keyboard, write to some screen or LED or serial interface.
Only when you want to run multiple programs, or deal with interrupts from outside,
conflicting ressources or such, then you will desperately need an OS.
2nd: The kernel is compiled to machine code, which executed during boot. There's no
C involved when running a kernel. C only helps writing a kernel or any program
which should run, if in the kernel or "bare metal".