You're talking some large distances OP. Google Maps can tell you how far it is between different places and how long it will take you to drive. I suggest you spend bit of time on there figuring out how long you'll be driving and whether those sort of distances in summer temps (around 100 degrees in the desert), are something you and your family will want to do day after day.
But, broadly, yes it's doable and many of those places are ones I've been to and are well worth visiting. However, if you want to hike in the NPs you'll need to get up early and get your hikes done before it gets too hot and carry lots of water.
Yes, you could fly to LA and out of Vegas with several airlines from the UK (BA, Virgin and their US counterparts). A friend of mine did two RV trips and said they were great, but RVs are huge, the roads in the NPs will be very busy in summer and you/your DP will need to be confidant that you can drive the thing. My friend and her DH have a camper van so they were confident driving a large vehicle, but an RV is fucking massive!
In terms of theme parks in LA, Disneyland is in Anaheim, an area south of LA in Orange County, as is Knott's Berry Farm (big roller coasters). Universal Studios is in Universal City, which is north of central LA and Warner Bros studios is in Burbank (also north of LA). Do your homework as to which will appeal to your family. We went to Warner Bros studios last summer and the focus is very much on old shows like Friends, Gilmore Girls, etc, which your kids may not be at all interested in (mine weren't, aged 14 and 11).