Is there a locker room or some place to leave some stuff through the day?

You can leave your luggage at the Amtrak parcel check. You do NOT need an Amtrak ticket. If you do, it's $10 per bag, if you don't $20.

Yes there is a lockers area just go to the help desk and they will escort you there and give you prices

Nope. Unfortunately, There's nowhere else locker rooms! They're removed it since in 1990s, You have to responsibility carried of your own stuff or LAPD will checks it if you left them while unattended