NCLEX inquiry

The answer to your question is what state do you wish to immediately start to look for your first nursing job. Then, go to that state's BON website (Google is your friend) and look at the educational and other licensing requirements.

The second question is assume you pass the NCLEX-RN, what sort of legal work authorization will you have to start that job hunt?

A valid SS# that's work authorized or a green card or working visa, say under a spousal or family process. If the latter, have you started the visa process, as you may or may not know, the current time frame is now several years before they (US) will begin to even touch and look at your application.

Many states will also require the passing of certain English proficiency exam(s), so again, you need to narrow down your focus on the state you wish to quickly look for a job, once you can clear the above comments made.

You don't automatically apply for NCLEX. You apply to a board of nursing for licensing by examination as an IEN. NCLEX is but one step in the process. All states are required to follow the federal regulations that no professional license is to be issued without a valid SSN, some states are slower than others to apply the rule.