Contact customer service, this sounds like CIS made a mistake.

The fact that you applied suggests you might want to immigrate to the U.S. and the J visa is a nonimmigrant visa. Immigrant intent could...

No, your naturalization should not disqualify them.

You are considered in status during the grace period after your F1 ends so yes, you can file for adjustment.

Marriage will not disqualify her from L or H visas.

It seems you filed prematurely and no doubt CIS rejected the application. Get a lawyer and refile.

Contact the Social Security Administration and file for one.

The local processing time at the new location will now apply to you.

The USCIS officer decides whether you are eligible, including whether you meet the good moral caracter requirements, for naturalization. Not sure...

You could try, but it would be quite risky. You may want to try consular processing for the H4 instead.