The LLC owns the domain, but one question is if you push the domain name more or less than the company name. After all, you don't go to Reckitt Benckiser’s web site to find out more about Calgon, Woolite, Clearasil, or Lysol.
2. If you put a copyright notice on your Web pages (a mouseprint copyright notice is normally a good idea for evidentiary purposes, although technically not required under U.S. law), then the copyright notice should use the official name of the LLC, not the site URL.
Usual disclaimers: I'm a lawyer, but not YOUR lawyer, so this isn't legal advice, don't rely on it as such, don't disclose anything confidential in the comments (lest you waive any privilege that might apply), you and I aren't establishing an attorney-client relationship via this thread, etc., etc.
In your hypothetical example -- assuming CoolCorp LLC was indeed the copyright owner -- a statutory copyright notice would be "Copyright © 2010 CoolCorp LLC"; there'd be no need to include the URL or name of the site.
I _have_ seen notices that say something like, "NewSite Copyright © 2010 CoolCorp LLC"; I'm not aware that adding the NewSite name like that would cause any problems.
Same disclaimers apply - I'm not your lawyer, etc.