The trace bitmap tool doesn't know what the "object" structure is of a bitmap, all it sees is contiguous areas of one color which it converts to closed, color-filled paths.
So the vector picture it generates will just be one object made up of paths. You can edit these paths with the node editing tool (second in the tool palette) by moving, editing, or deleting the nodes.
To separate the picture into different components, you can duplicate the object, move the duplicate to a different layer, delete everything except a selected component in the duplicate, and then delete just that component itself (the complement) from the original object, effectively splitting the object between two layers. And then do that recursively for each sub-component.