B767-200 CREW WAS GIVEN A LAST MIN RWY CHANGE AND INAPPROPRIATE VECTORS; THAT NECESSITATED A GAR IN WX AT JFK.

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WX AT JFK FLUCTUATED BTWN 1/4 MI VISIBILITY WITH STRONG THUNDERSTORM ACTIVITY AND 1 MI VISIBILITY IN FOG. AN ILS RWY 4R CAT III APCH HAD BEEN BRIEFED BASED ON THE MOST CURRENT ATIS. LATE IN THE VECTORING; ATC RPTED THE MOST CURRENT RVR'S AND REASSIGNED US TO ILS RWY 4L. WE WERE THEN ASSIGNED A TIGHT BASE TURN AT 4000 FT AND ASSUMED THE CTLR HAD ANOTHER VECTOR IN MIND DUE TO OUR EXTREME ALT. HE THEN DIRECTED AN UNEXPECTED DOGLEG TURN ONLY 3.5 MI FROM THE FAF AND TO MAINTAIN 2500 FT UNTIL ESTABLISHED; A 1000 FT ABOVE THE FAF ALT; WHILE AVOIDING TSTMS AND A 40 KT TAILWIND ON FINAL. THE ACFT WAS CONFIGURED; FINAL DSCNT CHKLIST COMPLETED AND WE CROSSED THE FAF APPROX 1000 FT HIGH. ATC HANDED US OFF TO TWR INSIDE THE FAF AT WHICH TIME IT BECAME OBVIOUS THAT A STABILIZED APCH WOULD NOT BE GUARANTEED BY 1000 FT AFE. AFTER CHKING IN WITH TWR; WE NOTIFIED HIM OF OUR GAR AND WERE DIRECTED TO EXPEDITE OUR CLB TO 3000 FT. THE ACFT WAS CONFIGURED FOR THE GAR BUT THE CAPT NOTICED THE SLOW CLB PERFORMANCE AND REALIZED THE SPD BRAKE WAS STILL EXTENDED. WE WERE THEN GIVEN A 90 DEG CLBING TURN; WHEN I NOTICED THE FLAPS WERE NOT CONFIGURED CORRECTLY FOR THE RAPIDLY ACCELERATING SPD. WE BELIEVE WE EXCEEDED THE FLAPS 20 DEG LIMIT SPD BY AN UNCERTAIN SPD; BUT VISUALLY CONFIRMED THE FLAPS WERE FULLY RETRACTED BY 230 KIAS. WE WERE THEN VECTORED BACK TO FOR AN ILS RWY 4R APCH. WHILE ON AN ASSIGNED HDG OF 310 DEGS NEARING THE FINAL APCH COURSE; THE SAME CTLR GAVE US A R TURN TO 300 DEGS. AFTER RESOLVING THIS ISSUE; WE WERE ON FINAL AND CLOSE TO A SECOND GAR DUE TO SPACING ON PRECEDING TFC. SEVERAL FACTORS CONTRIBUTED TO THE RESULTING SIT: A LATE RWY CHANGE; SHORT AND UNREALISTIC VECTORS/ALTS IN A TSTM/LOW VISIBILITY ENVIRONMENT AND CHOOSING TO DELAY OUR DECISION TO GO AROUND. THE SINGLE MOST PREVENTABLE FACTOR WOULD HAVE BEEN REALISTIC APCH VECTORS AND ALTS FROM THE CTLR. FROM THIS POINT ON; I WOULD NEVER ACCEPT APCH CLRNC OR A TWR HDOF WITH SUCH A POORLY DIRECTED APCH IN SUCH WX CONDITIONS.