Contact the state board of nursing you plan on working in- every state is different, some states take a long time to get a license in, other states are "walk-through" states, meaning you can go to the BON office & apply for & get your license in the same day, or within a few days via mail.
As far as flying or driving, I drive to my assignments, then I can bring more stuff with me, and I don't have to worry about renting a car while I'm on assignment- it all depends on personal preference.
For housing, I am not a home owner either, and I chose NOT to keep my apartment in my home state. Now, because of this, I do not have a "tax home" for tax purposes, so anything my company spends on me for housing in considered taxable income, so I do have some extra taxes taken off of each check, but it's still way cheaper than paying my own rent. I use a family member's address as my permanent address for mail, and have them mail everything to me once a week or so, and have all my bills, etc sent to that address. BUT, for tax purposes, I do not have a permanent home.