Notice: Undefined variable: node in eval() and Notice: Trying to get property of non-object in eval()
Something is trying to run the php_eval() function in php.module. Most likely you have something configured in your site (such as a block) to use the PHP input filter, which evaluates PHP code and prints the output as text. I would check all the blocks you have defined first.

You can also temporarily put a call to debug_print_backtrace() at the top of the php_eval() function and print it's results while browsing your site as an anonymous user to see what exactly is calling that function.