Is it possible to do the same with a fixed camera and rotating subject?
Yes. Moving camera or moving scene is essentially the same thing. It's just a matter of reference.
The only thing I'm not sure is if the changing background will be a problem. Will I have to recognize and extract the object I want to reconstruct to eliminate noise maybe?
Yes, a non-static scene is more problematic. You can solve this by detecting which parts are static, and which are dynamic, and then only use the static objects. It is however not trivial.