How do Pressflow Drupal sites handle user Cookies?
Cookies/sessions are handled by both Drupal and PHP. Drupal register some functions that PHP will call, when certain things need to happen, like destroying a users session. PHP handles most of the basic stuff and Drupal override some of it, store users sessions in the database.

I'm not familiar with Omniture but if it's anything like GA, a users session/cookie wont effect tracking at all. Cookies are used to identify the user, so they don't have to type in their username/password at every page. Without cookies, Drupal wouldn't be able to tell the users apart and would need some other form of identification, like a login for every new page load.

This is why, some people try to steal cookies, as that is the same as having the user log in and leave the computer, given the thief full access.