Can I take an existing query and run it against a different universe without having to build it again from scratch?

Yes. You can move queries between universes, which allows you to preserve all the query structure and the reporting formatting.  To do this, use the Properties tab in Edit Query mode and click the ... button (see Figure 1) next to the universe name to select from a list of all available universes. Then, you can use the Change Source window (see Figures 2 & 4) to map your query to different objects, if you choose, or remove them from the query all together. 

NOTE: If there are multiple queries from the same database in the report, it will attempt to repoint all of them to the same universe!

Figure 1. Location of button to click on the Properties tab to open the Universe dialog box in order to alter the mapping of objects in your query.

 

Figure 2. Change Source box where you select the taget object to change.

 

Figure 3. Map Object box showing choices to Remove object or Select a new object.

Figure 4. Change Source box showing changed mapping.