HIGH ON THE APCH; B757-200 INITIATES GAR AND ENCOUNTERS TCAS RA ALMOST SIMULTANEOUSLY. FLAP OVERSPEED MAY HAVE OCCURRED DURING COMBINED MANEUVERING FOR THE TWO PROCEDURES. RESULTING FLAP AND SLAT DISAGREE LIGHTS RESULT IN AN ABNORMAL FLAPS 20 LNDG.

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ZZZ APCH CTL HAD US ON THE VISUAL TO RWY XXL. THEY INFORMED US OF TFC LNDG ON RWY XXR; A B767; AT OUR 2 O'CLOCK POS 6 MI AHEAD. WE WERE TO STAY AHEAD OF THAT TFC; WHICH WE WERE ALREADY BEHIND. FO STARTED THE DSCNT AS WELL AS SLOWING TO TARGET SPD IN ORDER TO BE IN A CTLED DSCNT TO LAND ON RWY XXL. AS WE CONVERGED WITH TFC WE WERE STILL HIGH ON THE APCH; THEN WERE TOLD TO CONTACT ZZZ TWR. WE DID SO AND RECEIVED CLRNC TO LAND ON RWY XXL AND TO BE AWARE OF TFC NOW AT OUR 3 O'CLOCK POS; BUT STILL AHEAD OF US. FO WAS ON THE LOC; BUT STILL 1 1/2 DOTS HIGH ON THE GS. AT APPROX 1700 FT; TCAS ISSUED AN RA TO CLB. THE CAPT ORDERED A GAR; THE FO INITIATED ONE. GAR SWITCHES WERE ACTIVATED; GA APPEARED ON ATTITUDE INDICATOR AS WELL AS TOP EICAS PAGE; FLAPS WERE RETRACTED TO 20 DEGS; PITCHED TO COMMAND BARS; AT POSITIVE CLB CALLOUT; GEAR WAS RETRACTED. SPD CAME UP INCREDIBLY FAST; AND FLAPS EXTENDED SPD WAS EXCEEDED BY APPROX 20 KTS. PITCH WAS INCREASED AS WELL AS PWR WAS REDUCED TO SLOW SPD. BOTH LEADING EDGE AND TRAILING EDGE DISAGREEMENT EICAS LIGHTS AS WELL AS LEADING EDGE; TRAILING EDGE LIGHTS ILLUMINATED. ONCE AT LEVEL FLT; CAPT TOOK OVER AS PF; CALLED FOR APPROPRIATE CHKLISTS. CHKLISTS WERE COMPLETED; LIGHTS EXTINGUISHED; LNDG WAS SUCCESSFULLY ACCOMPLISHED WITH FLAPS AT 20 DEGS. MAINT MSGS WERE SENT; MAINT MET ACFT AND VERBALLY INFORMED BY CAPT OF INCIDENT. TWR GAVE US A LEVELOFF ALT OF 3000 FT AND A HDG OF 280 DEGS. THE ACFT RESPONDED RAPIDLY AND THE AIRSPD CAME UP QUICKLY TOWARDS 180 KTS; BUT AT NO TIME DID WE EXCEED 180 KTS IN THE GAR MANEUVER.