China Eastern is MU, not CE. CE is a non-existant IATA code at the moment.

Whether you need to re-check in depends on the airline you're originating with to get to LAX. If you're arriving on the same ticket from DL, then you should be ok and the DL agent would check you all the way through. If you're on separate tickets then it could raise an issue. I'd check with the Skyteam lounge when you arrive assuming you have access, otherwise check at the gate as soon as you're able. There's usually no hassle in the US doing your verification at the gate, it's a lot simpler than China departures.