A B767-300 AT CRUISE; FL360; RECEIVES A GPWS FROM KNOWN TFC BELOW AT FL350. SINCE THE TCASII HAD BEEN TURNED OFF DUE TO 'NUISANCE TA WARNINGS;' THE FLC CLBS OUT OF FL360. IN THE PROCESS OF ATTEMPTING TO USE THE GPWS GEAR OVERRIDE SWITCH; THE PIC DROPS THE LNDG GEAR. CREW WANTS THE NORTH ATLANTIC REDUCED VERT SEPARATION MINIMUMS PROCS TO BE MODIFIED TO PRECLUDE A REPEAT OF THIS EVENT.

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OVER THE ATLANTIC ENRTE MXP (MILAN) TO IAD (DULLES) B767- 300 AT FL360. ANOTHER ACFT FLYING BELOW AT FL350 REMAINED DIRECTLY BENEATH OUR ACFT FOR AN EXTENDED PERIOD OF TIME (30+ MINS). THOUGH WE COULD NOT SEE HIM VISUALLY; WE COULD SEE THE TCASII TARGET. WE RECEIVED REPEATED AND FREQUENT TA AURAL WARNINGS 'TFC; TFC.' THEN THE GPWS ALARM BEGAN SOUNDING REPEATEDLY 'TOO LOW TERRAIN; TOO LOW TERRAIN...ETC.' AS A RESULT OF OUR GPWS TRAINING AND FEAR THE ACFT BELOW MIGHT BE CLBING INTO US; WE IMMEDIATELY PULLED THE NOSE UP INTO A CLB. WE KNEW FROM THE PREVIOUS HRS OF TCASII OBSERVATION THAT THERE WAS NO TFC ABOVE AT FL370. WHILE ATTEMPTING TO INHIBIT THIS LOUD CONTINUOUS DISTRACTING AURAL WARNING; I INADVERTENTLY ACTIVATED THE ALTERNATE LNDG GEAR EXTENSION SWITCH WHILE TRYING TO ACTIVATE THE GPWS GEAR OVERRIDE. THERE WAS THEN A VERY LOUD ROARING NOISE IN THE COCKPIT FURTHER AGGRAVATING OUR ANXIETY. I THOUGHT MOMENTARILY WE MIGHT HAVE ACTUALLY MADE CONTACT WITH THE OTHER ACFT. THE GEAR EXTENDING ALSO CAUSED A NOSE DOWN PITCH WHICH FURTHER STIMULATED US TO APPLY YOKE BACK PRESSURE. PRIOR TO COMPLETE GEAR EXTENSION; I REALIZED MY ERROR AND I RAISED THE GEAR WITH THE GEAR LEVER. WE HAD CLBED TO APPROX FL368 AND SINCE WE COULD NOW SEE THE ACFT BELOW WE RETURNED IMMEDIATELY TO FL360 AND PROCEEDED TO DEST WITHOUT FURTHER INCIDENT. THERE WERE NO INJURIES OR DAMAGE CAUSED BY THIS INCIDENT. I BELIEVE PROCS SHOULD BE DEVELOPED TO PREVENT THIS TYPE OF GPWS ALARM ACTIVATION WITH THE NEW 1000 FT VERT SEPARATION. TO MY KNOWLEDGE THIS HAS NEVER OCCURRED BEFORE AND IS A MOST DISCONCERTING EVENT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE CAPT SAID THAT HE HAD FILED A COMPANY RPT AND HAD SUGGESTED THAT THEY INCLUDE THE EVENT IN THEIR NEXT SAFETY PUB FOR INFO DISSEMINATION. IN RETROSPECT; HE SAID THAT HE WOULD CONSIDER AN OFFSET MANEUVER IN THE FUTURE. THE REASON FOR HIS NOT USING THAT PROC DURING THIS INCIDENT WAS THAT HE HAD BELIEVED THAT THE OFFSET MANEUVER WAS TO BE USED ONLY FOR WAKE TURB AVOIDANCE FROM A PRECEDING ACFT NEAR HIS FLT LEVEL. HIS COMPANY DOES HAVE A POLICY OF NOT TURNING OFF THE TCASII INFLT. HE HAD NOT ESTABLISHED 2 WAY COMS ON 131.8 WITH THE OTHER ACFT BUT WILL IN THE FUTURE. BOTH ACFT WERE AT CRUISE MACH .80. AT THE TIME OF THE GPWS ALERT HE REALLY FELT THAT THE OTHER ACFT MIGHT BE INITIATING A CLB. WITH THE TCASII TURNED OFF HE FELT THAT HE HAD REDUCED HIS OPTIONS QUITE A BIT SO THEREFORE STARTED A CLB. SUPPLEMENTAL INFO FROM ACN 412067: AN ACFT WAS DIRECTLY BELOW US AT FL350; SAME SPD. WE TURNED TCASII OFF DUE TO CONTINUOUS NUISANCE WARNINGS 'TFC; TFC.' WE THEN HAD LOUD AND CONTINUOUS GPWS WARNINGS; 'TOO LOW TERRAIN; TOO LOW....' THE CAPT (PF) AND I IMMEDIATELY GRABBED THE CTLS AND CHKED THE RADAR ALTIMETER THINKING THE ACFT BELOW US WAS CLBING INTO US. WE STARTED A SHALLOW CLB. I THEN WENT TO THE OFFSET MODE OF THE FMC RTE PAGE. THE CAPT UNINTENTIONALLY MOVED THE ALTERNATE GEAR EXTENSION SWITCH; A NEGATIVE XFER FROM B727 SWITCH. POSTFLT INSPECTION BY MAINT REVEALED NO DAMAGE TO THE ACFT. THERE WERE NO RPTED INJURIES FROM THE CABIN. 1000 FT SPACING ON NAT'S NEEDS TO BE MODIFIED WITH PROCS LIKE TIMING OR OFFSETS. THE CURRENT PROC IS DANGEROUS. WE SHOULD NOT BE TURNING OFF TCASII OR GETTING GPWS WARNINGS FROM OTHER ACFT. SUPPLEMENTAL INFO FROM ACN 412226: I WAS CURRENTLY ON MY SCHEDULED BREAK (3 PLT CREW); SITTING IN PAX SEAT RESERVED FOR BREAK PLT. THE ACFT BEGAN VIBRATING/RUMBLING NOTICEABLY AND HAD THE SENSATION OF A PITCH-UP. I RETURNED TO THE COCKPIT; OBSERVED WE WERE AT FL360 AND ON FMS COURSE. I DETERMINED FROM THE CAPT THAT NO ASSISTANCE WAS NEEDED AND RETURNED TO MY BREAK.