Where is the OS when a process is running?
The question is mixing up who's in control of the memory and who's in control of the CPU. The wording “running” is imprecise: on a single CPU, a single task is running at any given time in the sense that the processor is executing its instructions; but many tasks are executing in the sense that their state is stored in memory and their execution can resume at any time.

While a process is executing on the CPU, the kernel is not executing. Its state is saved in memory. The execution of the kernel can resume:


if the process code makes a jump into kernel code — this is called a system call.
if an interrupt occurs.


If the operating system provides preemptive multitasking, it will schedule an interrupt to happen after an interval of time (called a time slice). On a non-preemptive operating system, the process will run forever if it doesn't yield the CPU. See How do modern operating systems like linux handle multitasking? for an explanation of how preemption works.

Tasks such as process management and device management are triggered by some event. If the event is a request by the process, the request will take the form of a system call, which executes kernel code. If the event is triggered from hardware, it will take the form of an interrupt, which executes kernel code.

(Note: in this answer, I use “CPU” and “processor” synonymously, to mean a single execution thread: a single core, or whatever the hardware architecture is.)