MULTIPLE DISTRACTIONS INCLUDING LATE RWY CHANGE RESULTS IN E145 FLT CREW LEAVING THE PAVED SURFACE WHEN EXITING RWY 31L AT JFK.

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FLYING INTO BUSY JFK ARPT; 250 KTS OR LESS ASSIGNED BY ATC. APCHING JFK ATC CONTINUED TO SLOW ALL ACFT INBOUND. ASSIGNED APCH SPD WELL BEFORE IAF. CTLR COMPLAINING ABOUT RWY CHANGES. ALL PLTS NEED TO PICK UP NEW ATIS. CAPT PICKED UP NEW ATIS. LNDG RWY 22 NOW BUT OUR FLT ASSIGNED RWY 31L; PREVIOUSLY EXPECTED RWY 31R. REBRIEFED RWY. OVER TO TWR AND CTLR YELLING AT ALL ACFT. SEVERAL LNDGS AND TKOFS ON RWY 31L BEFORE I TOUCHED DOWN. DURING THIS TIME AT 2000 FT USING ILS EVEN THOUGH CLRED FOR VISUAL. TROUBLE SEEING EXACT MARKINGS OF THRESHOLD. SAID I SEE THE DISPLACED THRESHOLD LIGHTS AND CONFIRMED WITH ILS GS. NO OTHER LIGHTS ON; NO VASI; NO PAPI. I TOUCHED DOWN APPLIED BRAKES AND OPENED THRUST REVERSERS AND TWR CTLR YELLING 'ACR X; HAVE YOU BEEN LISTENING TO WHAT HAS BEEN HAPPENING HERE? I HAVE ACFT LNDG BEHIND AND ACFT ABOUT TO TAKE OFF! I NEED YOU TO EXIT MB. MB. IS ANYONE UNDERSTANDING;' ETC...HE KEPT GOING ON AND ON. MEANTIME CAPT SAID I HAVE THE CTLS AND I SAID I HAVE THE TOPS. I CALLED OUT 60 KTS. THRUST LEVERS OUT OF REVERSE CHKED. THEN I LOOKED TO THE R SIDE FOR THE TAXI EXITS LOOKING FOR MB; AND WAITING FOR CTLR TO STOP RANTING! I PRESSED PTT 'ACR X EXITING ON MB OVER TO GND.' SAW IN FRONT GRASS 'GRASS! GRASS!' CAPT STOPPED ON GRASS. CAPT IMMEDIATELY NOTIFIED TWR WE WERE ON GRASS BY TXWY MB. TWR CANCELED ALL LNDG ON RWY 31L. ACFT ON EDGE OF GRASS OPS NOTIFIED AND TUG SENT OUT TO PUSH BACK ACFT. CAPT SIMULTANEOUSLY COMMUNICATING WITH CABIN PAX AND FLT ATTENDANT AND I WITH GND. THIS IS TO THE BEST OF MY MEMORY OF WHAT HAPPENED. I BELIEVE THE MAIN CONTRIBUTING FACTOR WAS THE TWR CTLR YELLING WHILE DECELERATING ON RWY. EXTREMELY DISTRACTING! LACK OF DIRECT CLRNC CAUSING CONFUSION OF HOW CLOSE OTHER ACFT WERE TO US. NO LIGHTS AND IT WAS DUSK TURNING TO NIGHT QUICKLY. NO REFLECTORS ON EDGE OF GRASS AREA OR TAXI LEAD LIGHTS. LIGHTS SHOULD BE ON ALREADY. COMPLAIN ABOUT UNPROFESSIONAL CTLRS YELLING AT JFK AND BLOCKING THE RADIO FREQ FOR OTHERS TO COMMUNICATE CLR CONCISE RESPONSE AND DISTRACTING PLTS AT CRITICAL TIMES ON THE RWY. AS COPLT I SHOULD HAVE LOOKED BACK TOWARDS THE FRONT OF ACFT QUICKLY INSTEAD OF LOOKING TOWARDS R SIDE FOR A LONGER PERIOD OF TIME. BACK AND FORTH SCAN WOULD HAVE BEEN BEST. EVERYTHING HAPPENED QUICKLY DUE TO HIGH SPD TURNOFF.