1. French Office of Immigration and Integration. See here:
http://www.ofii.fr/
2. Mostly yes.
A residence permit does not confer any rights more than what is originally granted by the issuing country. For example, if you have a U.K. residence permit for studying, you can't use the permit and reside in France. That permit is simply good for entering the U.K. while you remain as a student.
You will still need your travel document (i.e. Passport) when you travel (unless your travel is within Schengen area or Common Travel Area).
Only citizens of EU member states can enjoy freedom of movement, regardless if they have proper residence permits or not. Even so, citizens of EU member states may still need to apply for a permit as a formality.