User always have one session at a time. Every time he logs in or out, new session is created.

Drupal 6 does not clean session table. I don't know if it is supposed to or not, but both my experience and simple Google search reveals it fails to do it.

So you have current session of an logged-in user, and outdated session data he left behind when he stopped to be anonymous.

If this really bothers you, consider moving sessions to APC or Memcached, it's much faster than database way, and unused data will be purged by the storage system even if Drupal will not trigger cleanup.

User always have one session at a time. Every time he logs in or out, new session is created.

Drupal 6 does not clean session table. I don't know if it is supposed to or not, but both my experience and simple Google search reveals it fails to do it.

So you have current session of an logged-in user, and outdated session data he left behind when he stopped to be anonymous.

If this really bothers you, consider moving sessions to APC or Memcached, it's much faster than database way, and unused data will be purged by the storage system even if Drupal will not trigger cleanup.

The user is redirected back to the homepage and is presented the login form again and is not logged in. Ironically, the user's name appears in the footer of the homepage as a logged in user.

This behaviour I experienced only once - when database was misconfigured, and it reported inserts to session table as successful, but next request failed to read freshly inserted data. Apparently it was database caching, but unable to fix it, we ended up with sessions in RAM, as noted above.