A B737 CREW GIVEN AN INCORRECT CLRNC AND SQUAWK WERE MISIDENTIFIED BY ATC AFTER DEP. THE COM DISTRACTION CAUSED THE CREW TO FAIL FULLY RETRACTING THE FLAPS AND TO EXCEED THE FLAP EXTEND SPD.

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AFTER A 10 MIN DELAY AT THE END OF THE RWY; WE WERE TOLD TO TAXI INTO POS AND HOLD AND WAS TOLD WE WOULD BE IN POS FOR 2 MINS. AFTER THE WAIT ON THE RWY; WE WERE CLRED FOR TKOF AND TOLD TO CLB TO 4000 FT AND TURN TO A HDG 170 DEGS. AFTER CLBING THROUGH 1000 FT AGL; THE PF ASKED FOR FLAPS 1 CLB THRUST AND I REPEATED AND MADE THE SETTINGS. AT THIS SAME TIME WE HEARD ATC DIRECT A REQUEST TO ANOTHER COMPANY FLT NUMBER THAT HIS XPONDER WAS NOT ON AND THAT SAME COMPANY ACFT REPLIED THAT HIS XPONDER WAS ON. BOTH THE PF AND MYSELF LOOKED DOWN TO SEE OUR XPONDER WAS IN STANDBY AND THE CAPT TURNED IT ON. A VERY SHORT TIME AFTER THIS; ATC CALLED BACK TO OUR FLT NUMBER AND TOLD US TO CALL DEP CTL. UPON CHK-IN WITH ATC WITH OUR PROPER CALL SIGN; HE REPLIED TO US BY CALLING US THE OTHER COMPANY'S FLT NUMBER; TOLD US WE HAD THE WRONG SQUAWK CODE AND TO CHANGE IT. AFTER I CLARIFIED WHO WE WERE AND WHAT FLT WE WERE; HE GOT CONFUSED AND HAD NO IDEA WHO OR WHERE WE WERE. BOTH THE CAPT AND I WERE TRYING TO EXPLAIN TO ATC WHAT OUR POS WAS AND WHERE WE WANTED TO GO. IN A VERY SHORT TIME WE LEVELED AT 4000 FT MSL AND THE CAPT WAS HAND FLYING THE ACFT WHEN HE NOTICED A TRIM LIGHT ILLUMINATE; AND AT THIS SAME TIME; I NOTICED THE FLAP HANDLE STILL SITTING AT POS 1. I TOLD THE CAPT ABOUT THE FLAP SETTING AND THEN MOVED THE FLAP HANDLE TO THE 'UP' POS. THE TRIM LIGHT EXTINGUISHED. AT THIS TIME WE WERE APPROX 30 KTS ABOVE THE MASI SPD LINE BUT BELOW THE MAX FLAP SPD OF 250 KTS. WE CONTINUED TO OUR DEST AND CALLED MAINT TO THE FLT DECK FOR A FLAP INSPECTION. WE WERE BOTH VERY FATIGUED FROM THE PREVIOUS DAY'S FLYING AND EVEN TALKED ABOUT THIS WHILE WE WAITED TO TAKE OFF. WE SHOULD HAVE BEEN MORE ATTENTIVE TO EACH OTHER'S ACTIONS AND FOLLOW STANDARD COMPANY PROCS WHEN PERFORMING OUR DUTIES.