I have a PSoC with debugging functionality ( I am sure other MCUs have similar functionality) which allows you to place the system in debug mode and you are then able to step through program execution, view register values, etc.. When in this configuration does the entire system halt and wait for you to "step through" the code? or are there more processes running in the background?
It depends on the MCU. Some (like TI's C2000 line) have a debug mode that allows the CPU to take interrupts while stepping through non-interrupt code. Timing and communication peripherals may have an option to continue working while the CPU is halted. But unless the documentation says otherwise or you've done some special configuration, you should assume that all peripheral activity will halt along with the CPU.