There is no reason why an H4 may not work abroad. In general, Most US laws are unenforceable outside of US.

You may write USCIS with supporting documents you are now USC petitioner. If unsuccessful, take you naturalization certificate or US passport to...

You must attend your interview. If you fail to appear, your I485 will be denied for failure to appear, not criminal conduct. If you attend the...

Provide additional detailed information about your application and immigration hsitory. Occasionally, cards are produced in error and causes...

It appears you have exhausted all available administrative remedies. Instead of having the indent to file the writ of mandamus, file one.

You must disclose the arrest to USCIS and submit certified arrest and disposition as well. Yes, anytime you travel abroad, the incident may come...

It basically means you have 60 days to find another employment.

It is not required but may help depending on your circumstances. For a detailed discussion and guidance, speak to an immigration lawyer.

That may be resolved at your green card interview. With supporting documents explain the issue to reviewing USCIS officer.

A case status inquiry specifically tells you at what stage of the process you are. It may or may not be biometrics examination.