No. You don't need to mark a job "Records Received" before completing or filing it. An unfinished checklist does block complete job, though.
Whether chase reminders stop depends on the optional conditions set on each reminder. For example, a reminder can be set to fire only while the job status is "Request records" or "Awaiting records", in which case once you're past that stage there's nothing to stop by updating this field. A reminder with the condition "Record receive date is not provided" stops once the records received date is set.
To check yours, go to Settings > Jobs > Services > Global auto reminders > set up and look at the Conditions column.
