Resources for backgrounds and objects for animations

Character Animator works very well with After Effects through Dynamic Link, you could very easily create a scene in AE and place your CH puppet into it.

Some sample backgrounds can be found here under Additional Resources. The backgrounds in the above link are single layer jpg files, you can make them as detailed or as simple as you like.
You'll also see a Physics Playground download under All Puppets. Which might give you some ideas about how a character can interact with content in a scene.

As Alan suggests you can have different items on different layers in CH so your character can appear to walk in front of some and behind others.

CH has no special support for what I think you refer to as a scene. Only puppets. For backgrounds, I just create a picture and drop that it. It becomes a puppet, but you only use Transform behavior to scale it to the right size and any minor position adjustments.

If you want furniture (e.g. a desk), you create an image layer with alpha channels (transparency). You put the furniture above the character above the background.

CH has no other special “scene” support, unless you are referring to the particle system (you can have simple bouncing balls etc - which again are just puppets with artwork like all the other puppets). The only primitives are puppets, audio files, and CH scenes (where you add puppets and audio files into). Very simple model.