If I have multiple (independent) checklists on a card, is it possible to change the order in which they appear? I know I can change the order of items within a checklist.
There is not currently a good way to change the order of checklists.
Checklists are arranged alphabetically by name... so you could prefix the name of the checklist with a number to control the ordering. You may have to refresh the page for the new ordering to take effect.