If a bunch of different characters were all doing the same dance, for example, is it possible to animate one and then copy/paste certain track items (camera input, mouse input, etc.) onto other characters (if they have the same anchor points)?
Not at the moment. The only copy/paste functionality offered by Character Animator is the ability to copy face track data from video in After Effects and paste it into Character Animator. For more information: After Effects Help | Face Tracking
Hope that helps.
While it is not possible to copy/paste recorded actions, it may be possible to record the same actions at one time on multiple puppets.
You can try this --
1. Place several puppets in the scene
2. Select all the puppets
3. Hit record and do your dance
4. Play back and see if all the puppets are dancing.
You can also try it for lip sync