Renter's Insurance is a confused situation IMHO. At least one of the well known brokers will take instructions from a foreigner and even put your correct foreign address on the documents that they email to you. However, they still insist on a US mailing address "to send the paperwork to". All rather odd. I wonder how much they pass on to the underwriters.
Regarding your planned flight, I strongly recommend that you make use of the free 1-800-wx-brief service to file your VFR flightplan, check on Notams and TFR's and get the weather, all from a human being who will (almost) always be extremely helpful.
Do get Foreflight on your iPad if you have one. Foreflight is different to UK offerings, much more intuitive and reliable. It also uses the real sectional chart and overlays real time weather.
I agree the comments on KLAS, especially as the FBO is on the 'wrong' side of the airport and the benefits you thought you were going to get, like free hotel shuttles, don't work out. Personally I always go to VGT which has a Hertz rental desk (phone ahead). If you get a car you don't need a hotel reservation and can shop around for prices on the day. There's a bank of free hotel courtesy phones in the lobby at VGT. A cab from either Henderson or VGT will be $30 - $50 anyway.
GCN is not the fearful place that some people describe but can be busy with tour traffic. Be sure of your DA calculations and expect the ground roll to be longer than you are used to. It's 7000' altitude but you've got 9,000' of runway. Resist the temptation to rotate early.
Consider an overnight stop at GCN because getting around the park on the shuttle bus takes longer than you'd expect and the afternoon turbulence may be accompanied by t-storms. There are modern hotels at Tusayan, the village next to the airport, within walking distance along a path through the woods (perhaps not with kids and baggage!). Expect to wait a long time at the airport for a cab. Maybe the hotels pick up?
Flagstaff is a lovely airport and the town now has modern hotels, at least one of which picks up. Don't miss the observatory. The kids will enjoy looking through the telescopes (for a while) but avoid the lectures! Another high altitude / early morning departure. Consider a side trip to Meteor crater and Sedona, again not the difficult place often described except perhaps in strong winds.
Lake Havasu is a gorgeous airport with a terrific FBO, Desert Skies. They will lend you a courtesy van and you will need it because it's 7 miles downtown. If you ask nicely they might even let you keep the van overnight. Don't get me started on courtesy car insurance! The London Bridge is interesting but I bet the kids won't think much of it and the 'London Village' is decrepit. The hotels are Ok and you can do waterskiing. Best part is flying along the Colorado river.
Enjoy your trip!