If I have a final hearing with Family Court to see if I've paid certain money to my ex but I've filed an appeal with the Supreme Court, is there a way to get a continuance from the Family Court pending the appeal?

Some courts require a formal motion for continuance-- however, others will reset on the agreement of the parties.  You need to call the court's coordinator and ask the judge's preference.  A formal motion for continuance is always good because it leaves a papertrail of what happened and why-- so filing a formal motion and listing the appeal a a basis is liklely to get your case reset until the appeal is perfected.  If you and the opposing side can just agree on a new date, then you can both communicate the agreement to the court coordinator.  However you get it worked out, make sure that your court's coordinator knows the status of the case.