I think you said you will be coming into LAX. I used to own an RV rental company so let me do a little research and see if I can find a decent rental place that meets your needs.
As far as TV - it depends. Most of the large rental companies, like cruise America, have rv's that are really stripped down. No slides, no awning, and usually no TV's - at all. They are bare bones. A smaller company or private rental will be a far nicer unit. Private rentals may also have it stocked with necessary items (pots, pans, dishes etc) We offered these things with our business. We sold our business a few years ago or I would be glad to set you up. But let me see what I can find.
If you have an RV with a TV, typically as long as they have an HDMI port, you can get a firestick and you will have internet TV on the go. The caveat is you need a data plan with a hotspot and you probably want unlimited data. This may be a problem since your trip is so short. I know folks on here know about pay as you go data plans, so perhaps someone can chime in.
There are other options, like portable DVD players, tablets loaded with movies, things like that. Normally, my advice would be to enjoy the scenery but realistically with kids that young you have to be pragmatic - they will need entertainment for long days on the road.
Bike rentals - you might have luck at a local bike shop near a destination. They may also have racks.
Licensing requirements - just about anything you rent will only require a standard license. It's not until you get into larger RV's over a certain weight limit, air brakes etc, that the requirements change. I don't know what your budget is, but you can find large luxury RV's to rent, like a Prevost. But that will be a tremendous amount of money. Like rock star money. I don't want to assume, so maybe you have that kind of wealth. If you do, let me know, that changes things and opens up some really luxurious coaches with all the frills you can imagine. These will limit where you can camp though as a lot of national parks have size limits.
If not, it really seems like a gas class A is what you are after. It may not check all your boxes, but everything is a compromise with RV ing. But I think Gas class A is your best bet. this will still be expensive - estimates for a nice gas Class A - will be in the neighborhood of $250-450 per day. Mind you - you will still pay close to that for the stripped down Class C at a major company, so a little extra goes a long way and will make your trip much more enjoyable. And that is just the RV. You still have fuel, LP, campgrounds etc.
I'll do some digging.
on edit:
Did some quick research for Class A Rv rentals in Los Angeles. There is a LOT to choose from.
Here is a site that caught my eye.
https://rvshare.com/usa/california/los-angeles/rv-rental/class-a
It has reviews and superficially prices look competitive. On par with the stripped down class C's I mentioned.
Just be very careful with the fine print. That price per day is the base, you will have fees so please make sure you account for those.
This looks nice, sleeps 8, has bunks etc. Diesel so will ride extremely well but still only 36 feet in length. Anyway, some food for thought. I estimated a trip and it comes in at close to $350/day which is a pretty good price for all it offers. It includes dishes a bike rack, TV's, DVD players, HDMI cables, etc. Again, just a cursory look, but this seems to check a lot of the boxes for you. It even states, "international travelers welcome", so they are likely used to helping you with your particular scenario.
https://rvshare.com/rvs/details/1005673
