Yeah. SF are the Green Berets. But now all, essentially, all fall under SOCOM. Each has their specialty...ie SF specializes in foreign internal defense, if I remember correctly.
Yes, except for the REAL "ninjas" in JSOC (CAG and ST6,among others) who's commander answers directly to the "National Command Authority", IE, the SecDef and Prez.
Foreign Internal Defense is the peacetime mission conducted in permissive or semi-permissive locales and the classic Special Forces mission is UW(Unconventional Warfare), which is basically FID behind enemy lines. IE 5th SF Group in the fall of 02 in Afghanistan. There were never more than 100men on the ground except when the Rangers and CAG hit a place up north.