Can the Appellate Court stay your District Court case?

Whether an appellate court can stay your district court case depends upon a couple of things.
First, have you asked the district court to stay the case?
In most cases, a party needs to request a stay of all other proceedings in the district court or other agency first. If the district court refuses to stay the case, then a party to an appeal or appellate petition can ask the court of appeals to stay the case. Please see rule 17 of the Utah Rules of Appellate Procedure for the appropriate process under rules 17.
Second, has the appellate court taken jurisdiction of your case?
You cannot expect the appellate court to stay your district court case unless you have filed an appellate document in the appellate court. In other words, you cannot bypass the district court and file a district court stay in the appellate court. This is a question of jurisdiction. The appellate court does not have jurisdiction to interfere in the lower court’s resolution of your case.
The one exception is when the request to stay is itself in the form of a petition for extraordinary relief under rule 19 of the Utah Rules of Appellate Procedure. In most cases, a petition for extraordinary relief is not a stay; rather a motion to stay the case–and perhaps for expedited review under rule 23C–would be filed alongside a petition for extraordinary relief. But a party may ask the appellate court to stay the case under rule 17 once the appellate court has obtained jurisdiction after the filing of a petition for extraordinary relief–or some other appellate document invoking the jurisdiction of the appellate court.
In many instances, a stay of judgment, as well as proceedings, will require a bond and attorney fees under rule 62 of the Utah Rules of Civil Procedure. However, if judgment is paid, only a nominal bond must be paid.