Hmmm, 15, 6 and 2. That must be tricky.
I fly back to LAX with my DD about twice a year, about half the time on my own. I always INSIST on going direct so if the Virgin flights didn't work out for me I would be driving to Vegas. Also, if you decide on San Diego British Airways having started flying direct to/from Heathrow, so you could go in to LAX and out of San Diego. I've never done Virgin with kids, but BA is great, particularly with Terminal 5. I don't fly with American carriers with my DD despite the fact that my father can get me tickets for £50 plus taxes. Snobby, I know but I did it once and never again.
Car hire is fairly cheap and petrol will be laughably cheap to someone from the UK. My experience is that the issue Brits have with driving in the US is the big distances, but I'm sure you know your limits. If you do plan to fly to LAX and then drive to Vegas you'll probably want to do the first flight of the day or the last flight, get a hotel room, have sleep and then head out really early. It should be about 6 hours and you have cross LA in it's entirety but then there's not much else until you get to Vegas.
ahh.. I'd skip it but that's purely my personal opinion.
If I remember correctly it another 6 hours to the Grand Canyon and what you end up doing there will probably be determined by whether you bring the kids or not. What you want to see/do at the Grand Canyon?
I assume when you say Anaheim/ San Diego you mean Disneyland or San Diego? I would say that the 6 y.o. would love Disneyland but the 15 y.o. is too old and the 2 y.o. is too young. If you take the kids I would recommend San Diego (disclaimer- I am from San Diego) You've got the SD Zoo which is very good, Sea World, which is obnoxious but worth it once. And you have Legoland, which is great for the little ones. The older one could hang out at the beach or do surfing lessons, if that's the sort of thing they are interested in.
Of course, I would suggest totally different things if the kids weren't going, but I think America is pretty easy to travel with kids. There's a lot of space and things like restaurants and hotels are easy. Having a shared language would make it easier for the older one to be a bit independent as well.
Oh, and I would do 10 nights, it's way too much internal travel for 7 nights.
Feel free to ask more questions.