Hope you had a good flight
It is not a requirement to have car insurance here but obviously it is beneficial in case of accident/theft and you're not having to shell out all over again. Other than that you can drive for up to 12 months on your US licence and you won't have to take a test to get a Kiwi one (will cost you about $60 to convert).
Shado Lans is great. Our pooch spent 30 days there under the old rules. She was a bit of a porker when she came out though - a mix of no exercise and feeding normal portions. But the facilities are great and visiting is easy.
We did a day room at LAX Travelodge before. It's easy enough, cheap and the rooms aren't all bad. We got the bus to Manhattan Beach from right outside, had a wander and some lunch. I realise it's too late for you now but might be useful for others
Always up for a coffee. We live in Island Bay but I work in the CBD.