Perhaps "Article III court"?

The judges are listed on the court's website: http://www.fisc.uscourts.gov/current-membership. They're regular U.S. District Court judges appointed to the court for a specified term.

The judicial Power of the United States, shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish. The Judges, both of the supreme and inferior Courts, shall hold their Offices during good Behavior, and shall, at stated Times, receive for their Services a Compensation which shall not be diminished during their Continuance in Office.

Congress has an authority to create courts as they see fit, now FISA might not be perfect it might be a complete disaster, but it's not unique as all modern courts have issues on being completely transparent at all times.

For example court orders such as search and seizure, peak and enter, wiretap etc' cant be public for obvious reasons since you can't argue that when you need to tap some one's phone to gather enough evidence to convict them of a crime you can make it public. You could also argue that such warrants violate the whole "innocent until proven guilty" posture by simply issuing a warrant before there's actual evidence of that a crime has been committed, or in fact it allows the state to force people to incriminate them selves in future crimes.

Now back to the FISA court the hearings and it's orders on specific efforts against specific targets can't be public, some oversight and process transparency might be possible but you can't make a court like that public.

That said the US is pretty much the only country which actually has a court like that, everyone spies, not to the extent that the US does but no one else have those resources, but I haven't heared about any other country in which the intelligence services have not only executive/legislative oversights but also judicial.