For the favicon issue see http://drupal.org/node/174940 which is looking to be ported to D5 but is already taken care of in versions higher than that. You can, also, see http://drupal.org/project/favicon. Quoting Earnie Boyd <earnie@users.sourceforge.net>:
Quoting Larry Garfield <larry@garfieldtech.com>:
If your watchdog table is filling up with PHP warning messages, then you have a serious bug in your code somewhere. Do not hide the error. Fix the bug. :-) Look at the error message itself to see what the problem is, track it down, and squish it.
There's a few from DRUPAL-5 that happen with every page load such as "page not found 2008-08-08 14:56 favicon.ico". There details of the "warning" do not tell me a referrer. To squash it I tend to "touch favicon.ico" in the DocumentRoot directory.
Earnie -- http://for-my-kids.com/ -- http://give-me-an-offer.com/