How fun!!

We took our DD when she was 19 months to Japan and the USA. We paid extra for a seat for her on the plane and took her car seat. It was fine in our rental in the USA. LA is really spread out and public transport isn't fantastic so you really need a hire car. There are a couple of rental places at the airport and others are a bit further away but have free shuttles. It's a beautiful drive through the desert to Arizona, maybe you can do a road trip?

Have a great time!!