Each individual justice may reject the cases submitted to him/her. All of the NBC cases were first denied/rejected by an individual justice.
The law allows the petitioner to resubmit to the justice of his/her choice. If that justice denies/rejects the case, the petitioner may continue until all 9 justices have denied the case.
Some aruge that these cases were sent to conference for that reason - to prevent the petitioner from exhausting all 9 decisions. (The case goes to conference where all 9 vote to deny/accept.)
I believe this is the procedure for stays.
What you are describing is the procedure for motions, under Supreme Court Rule 22. A request for the Court to hear a case is made via a petition for certiorari, which is always acted upon by the full Court, under Supreme Court Rule 15.5.