No, they are not proper nouns.
You capitalize Judge Smith. You do not need to capitalize retired as it is not part of his title.
You only capitalize it if it is part of a proper noun. The FBI is the Federal Bureau of Investigation.
no
Only when his name follows
Yes, It should be Federal Centers for Medicaid.
Yes. It should be--- Federal Work- Study Program.
A Federal District Judge has a lifetime appointment.
The judge retires.
Actually, it is acceptable to use "Judge (Name of Judge):" The rule is that you always capitalize a person's title.
Only when you are referring to a specifically named judge that you have identified in your writing. When referring to 'a judge' in the abstract, or 'judges' collectively, it can be in lower case.
Yes, "Federal Prosecutor" should be capitalized when referring to the official title of the position.
When it is used as a title the n it should be capitalized. Letter to the Judge