Second problem: Users cannot remain logged out. They log out, return to their journal, and are automatically logged back in.
http://www.livejournal.com/support/see_request.bml?id=54351
http://www.livejournal.com/support/see_request.bml?id=54333 (the request that makes me think this isn't cache problems)
http://www.livejournal.com/support/see_request.bml?id=54327
http://www.livejournal.com/support/see_request.bml?id=54077
I know that at least the second problem might stem from the recent changes to how cookies are handled; does anyone have any ideas on how to fix this?