I recently took the metro down to Long Beach from Hollywood and it was very easy. The price is $1.25 each way. You can also buy metro passes at the metro station which might be a better deal for you and I strongly recommend. I also recently took the train down to San Diego from Los Angeles and it was $35.00 each way and also very easy. You'll need to take the metro from LAX Airport to Union Station to get the Amtrak train that will take you to San Diego. Price for all Metro rides is again $1.25. I think that you will be fine without a car in both Los Angeles and San Diego. San Diego has excellent public transportation and Los Angeles' public transportation has greatly improved and I have started to use it more. You will be able to access all tourist areas of the city using either metros or buses. Beverly Hills, Venice, Santa Monica will be accessible by buses only, while Hollywood and Downtown have numerous Metro stations.
You won't find hostels in LA the way you find them in Europe, especially near the airport, however you can probably find a very cheap motel which will be near LAX and that will give you peace of mind to be at your flight on time for your return.
If you decide to stay all three nights on the QM, then the Hotel QM will arrange for a taxi to pick you up early in the morning and can also request a taxi that will take a credit card. The price to LAX from the QM will probably be about $40.00 plus tip. The ride will take about 30 minutes without traffic. I would not recommend using a metro or a bus to get to the airport for 8:30. I would only take a taxi for this since you have a flight to catch.
As for tours, once you book your stay at the Hotel Queen Mary, the concierge should be able to book a tour that will pick you up at the QM and return you there. If not, there will be a meeting point such as the Long Beach Aquarium, etc. which is nearby. The Queen Mary is a major tourist attraction/hub in LA/Long Beach, so there should be lots of services and tours right there at the QM.
