FLC OF AN LGT ACR ACFT ROTATED EARLY DURING TKOF CAUSING A TAIL SKID STRIKE; DAMAGING THE TAIL SKID.

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DURING TAXI TO RWY 35R AT DEN; I WAS PREOCCUPIED WITH COMPUTING THE MAX TAILWIND COMPONENT WE COULD ACCEPT. THE FLT WAS RWY LIMITED DUE TO TEMP. THE ATIS WIND WAS 130/04. LOAD PLANNER HAD NOT APPLIED TAILWIND CORRECTION. CAPT CALLED FOR TAXI CHKLIST AND WHILE HANDING THEM THE WT/SPD CARDS; I DROPPED THEM BEHIND THE TRASH BAG BTWN THE CONSOLE AND THE FO'S SEAT. I RETRIEVED THEM AND TOSSED THEM OVER THE THROTTLE QUADRANT AND ON TOP OF THE RADAR SCREEN. THE TKOF ROLL WAS NORMAL; AS WAS ROTATION; BUT THE ACFT WAS VERY SLUGGISH LEAVING THE GND; SETTLING BACK TO THE RWY ONCE BEFORE BEGINNING THE CLBOUT. AFTER THE PWR WAS PULLED BACK TO CLB SETTING WE ALL SAW THE WT/SPD CARDS WERE SET TO 144000 LBS. ACTUAL WT WAS PLUS 156000 LBS. INCORRECT BUG SETTINGS HAD CAUSED US TO ROTATE EARLY. WE HIT THE TAIL SKID; COMPRESSING THE CARTRIDGE ABOUT AN INCH. WHILE AFT SHUTTING OFF WATER TO AFT LAVS; I NOTED NO UNUSUAL NOISE OR VIBRATION. THE TAIL SKID INDICATOR WAS ON AFTER GEAR RETRACTION; BUT I BELIEVE THE TAIL SKID WAS NOT FULLY RETRACTED RATHER THAN EXTENDED. FLT CONTINUED WITHOUT FURTHER INCIDENT. PRIMARY CAUSE WAS FE'S FAILURE TO VERIFY CORRECT WT ON WT/SPD CARDS. CONTRIBUTING FACTORS: EXCESSIVE PREOCCUPATION WITH TAILWIND CALCULATIONS; SOME DISTR FROM OVERHEAD SPEAKER; AND SOME FATIGUE FROM PREVIOUS DAY'S COMMUTE AND SLEEPING ON COUCH IN CREW LOUNGE PREVIOUS NIGHT.