1) You may be, inasmuch as the wording for the license may be covered by its own copyright, and your use of that wording may constitute an infringement. Of course, it's somewhat debatable whether or not a fairly vanilla, standard license would contain enough originality to actually warrant strong copyright protection, but in general it would be safer to look for licenses that are themselves licensed to allow copying or the creation of derivative works.
2) You can register a copyright in your website with the Copyright Office (copyright.gov), and would do so by submitting the registration along with a copy of your website. As far as filing your licenses or anything else, there's generally not any place to do so, although an actual assignment of your copyright would be filed with the Copyright Office. And in general, whether or not your business is registered is largely irrelevant to copyright law, although if you're not doing business in your own name, many states require you to register an Assumed Business Name or DBA in order to carry on business or maintain a lawsuit in the name of the company.