First night - go straight to Beverly Hilton. No need to stay near the airport.

Rental car - Normally, I would suggest you skip rental car for the first 2 nights that you are staying at Beverly Hilton. However, since your kids are small, you may have to carry a lot of stuff with you during the day, a rental car may not be a bad idea. Beverly Hilton is walking distance to Rodeo Drive (one of the place you want to visit), and is located next to a major transit hub (intersection of Wilshire Blvd and Santa Monica Blvd). You might end up paying for parking for 2 or 3 days (at $40 a day) and not even use your rental car very much. The 720 (Wilshire) and 704 (Santa Monica) buses are 2 of the most popular and frequent bus lines in LA and there is usually a bus every 3 or 4 minutes. The 720 especially will be very useful for you if you want to take the kids to visit the beach/pier in Santa Monica, or other kid-friendly attractions like La Brea tar pits, Petersen Automotive Museum or the Children's Museum.

JW Marriott - You can visit Hollywood area from JW Marriott by subway so your rental car may end up spending a lot of time in the garage. You should buy tickets for Lakers game ahead of time. If you find Lakers prices a bit expensive, you may want to consider catching a Clippers game (LA's other basketball team, which many here think is having a better year than the Lakers), or Kings game (LA's ice hockey team). All three teams play at the Staples Center, right next to your hotel. However, you'll need a car to visit Disneyland (see next).

Disneyland - You can stay in Beverly Hills or Downtown LA (JW Marriott) and just take a day trip to Disneyland. This is best done with a rental car. Leave your hotel around 9 AM after the worst commute traffic is over and drive down to Anaheim. You should get there in about 1 hour. Disneyland is not as big as people tend to think so you can cover the entire park in one day with plenty of time to spare. Your admission ticket also gives you entry to California Adventure. It is up to you if you want to hit up both parks but I think your kids will be tired after a few hours at Disneyland given their age. You can stay until after dark for the fireworks and then drive back to your hotel in LA, again after the worst of the commute traffic is over. DO NOT attempt the Disneyland trip on weekends... it will be a miserable time for all.

Las Vegas - Keep your rental car for the Las Vegas trip (I assume you are driving). The Hilton Grand is located a bit far from the rest of prime section of the Las Vegas Strip so you will end up driving to the big casinos at the southern end of the strip. Hilton Grand is family friendly hotel and a good choice for young kids.

Edit: Just wanted to add that since your AUS$ is at parity with US$, everything will be literally on sale here. Feel free to "shop til you drop" and help out our economy There are lots of shopping options both here in LA and on your way to Las Vegas.