Is it possible to copyright a partially completed work, so that you may post the progress on a personal website, also owned by you to show progress?

I think you misunderstand how the law of copyright works. First, "copyright" is a noun, not a verb. You obtain a copyright the minute your work of authorship is reduced to tangible form, whether it is complete or incomplete, a draft or a fragment. You can register your copyright with the United States Copyright Office to obtain certain additional enforcement rights, but you have no better or worse copyright than the one you had when you wrote your first draft. Since you subsequent finished work will be deemed a "derivative" work of the original draft, you are largely protected in the final version if you register an earlier draft. You will wish to consult with a copyright attorney about this. Generally speaking, characters and fictional places are not protected by copyright law. What copyright law protects is expression. That means, for example, that you are free to write a book about something that takes place in Narnia or Oz, or (subject to certain limitations) that contains a lead character with a name similar to that of another book, as long as your text is not substantially similar.