1) can they live in Arizona or other hot desert places?

You're in a great location if you're in the desert. Only thing better would be high humidity as well. If not already acclimated, introduce slowly to increasing Sun.

2) how big of a pot should I grow them in?

I grow them in large pots as babies, but many will tell you to keep the pot small and sized just larger than the plant. This will stunt their growth in my experience. (some people want small plants)

3) how much should I water them?

In growth, treat them like tropical plants and water away. When not in growth, withhold water and treat them like a cactus. I just try to keep mine in growth all year long, although I know some of my newer plants do go dormant. I use almost straight Miracle Gro potting soil for mine, but others will tell you that in their area they must use a "gritty mix" that's basically small rocks. I would not recommend your using the gritty mix in the desert, but you may decide otherwise.

Thank you.

Here is a blog I like on growing Adenium's. It is written by a GW member. http://kalachuchiatbp.blogspot.com/p/adenium-in-cold-places_8.html

Thanks.