How much money/hr should I expect?

What you should expect as an hourly rate will vary widely depending on where you go (for instance Florida is one of the lowest pay areas), the type of facility, the need for your specialty and most of all what the economy is doing!
You can expect to pay extra for the 2 bedroom - generally the one bedroom is standard and you pay the difference between the cost of that and the cost of a two bedroom.
As for a car, why not drive to your contract? It will be much less expensive than the cost of flying you and your family and then renting a vehicle. If you give yourself plenty of time to stop along the way road trips can be really enjoyable. With driving you can also take more "stuff" with you - which will make the transition easier for kids if they can take along their favorite toys, etc.

I would just drive. Just take your time. I drove from Alberta Canada to LA California. Me and my fiance drove 24 hrs straight from home to Vegas, stayed for a few days, then drove the rest of the way to LA. I wouldnt recomend doing that with kids of course, but make a road trip out of it. Overall, it took about 30 hrs of driving