Typically AA will require passengers to re-claim their bags if they have an overnight connection en route to the final destination. Personally - I prefer this since you will be in control of the bags and not the luggage looting airport workers (which are a very small minority of the AA employees).

Upon check-in tell the agent to only check your bags to LAX. This should not be an issue.

LAX - Hotels? I suggest you Price Line a 4* hotel in the LAX area. They all have shuttle service and the 4* ones are the descent ones. There are a few very budget level hotels in the area that I would avoid. If you book directly with the hotels you will pay about twice the PL rates. If you are not brand loyal this will save you quite a bit.