I’ve long wanted to know:

• how to move records from one collection to another in Zotero. Drag-and-dropping items creates a copy in the new collection.
• which collections hold a certain record?

And it turns out this functionality is present in Zotero, and documented on a support page.

To move records,

To move items between collections, hold down Cmd (Mac) or Shift (Windows/Linux) when dragging items to the new collection.

And to show which collections hold a record,

To see all the collections an item is in, select the item and then hold down the Option (Mac), Control (Windows), or Alt (Linux) key. Keep the key pressed. After a brief delay, the collections containing the item will be highlighted.