LA to Miami or New York

[Hornet]

Hello

I am planning a trip in the USA next year and need some advice please. The plan is to send my bike to LA (from Sydney - any idea on the cost of doing this?) and then to ride to either Miami or New York (via Miami). The questions I have are:
1. What is the best time of year to do this? I enjoy heat and don't mind travelling in the middle of summer. Would prefer to avoid winter and the rainy seasons.
2. How long will the trip take (and how far is it) to do at a reasonable pace - 300 - 500 km per day.
3. How much camping will I need to do? Would prefer to stay in budget motels / backpackers lodges if possible.
4. Can I go via Mexico and if so, what will this add in distance and time?
5. What would be a resonable budget for such a trip?
6. I am on a New Zealand passport - do I need a carnet for the USA and Mexico?

I plan to ride a BMW 650 GS - any thoughts on whether this is a suitable bike and if not, what else should I consider.

I would really be grateful for any help.

Many thanks

Peter