These operations should be done during off-peak hours.
In fact, the CICS should be stopped ONLY if running either TSS/CICS or ACF2/CICS but either way there is no need to shutdown CICS address spaces since the ENF/CICS hooks will remain in storage during the recycle of ENF. So there is no impact to CICS regions.
So only ENF can be recycled.
About the STC, in general, the only time to be wary of an ENF recycle is when using ENF to track events for the scheduler.
This is because the scheduling product should not potentially miss an job event while ENF is not active.