I set an alert every time I start a new session, but it never appears in my event log on the sidebar. Do alerts "expire" after a few hours of not being on the site? Is anyone else having issues with the alert system?
No, they should be there. They work for me, it's odd they don't for you. Is it just training alerts that don't show up, or others, like game alerts or pings?

Game alerts and pings still show up. It's just the training center.