STUDENT PLT OF A LONG X-COUNTRY TRAINING FLT LOST CTL OF THE ACFT ON ABORTED TKOF RESULTING IN THE ACFT GOING OFF THE SIDE OF THE RWY AND DAMAGING THE NOSE GEAR AND PROP.

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ON LONG STUDENT X-COUNTRY; DOOR POPPED OPEN AFTER TKOF. AS INSTRUCTED; I GAINED PATTERN ALT; LANDED; AND THEN SECURED THE DOOR. ON TKOF; JUST AFTER ROTATING AND AT ABOUT 10 FT AGL; THE DOOR POPPED OPEN AGAIN. I ABORTED THE TKOF AS I'D BEEN TOLD TO BY PULLING THE PWR BACK AND FLARING UNTIL THE STALL WARNING HORN WENT OFF. WHEN I'D LANDED ON THE MAIN GEAR; I LET THE NOSE DOWN AND IT SEEMED 'MUSHY;' CONTINUING TO FALL FORWARD. I PULLED BACK ON THE STICK; AND THE PLANE VEERED TO THE L AND SKIDDED ON GRAVEL AND WEEDS; HITTING A RIDGE OF DIRT WHICH DAMAGED THE NOSE GEAR AND PROPS. AS A STUDENT PLT; I HAD 1 DEMONSTRATED ABORTED TKOF AND 1 WHERE I FOLLOWED MY INSTRUCTOR'S MOTIONS ON THE CTLS. EMER PROC WERE DISCUSSED SOME; BUT NOT IN GREAT DETAIL; AND IT WAS NEVER MENTIONED HOW TO TURN FUEL TO 'OFF.' DENSITY ALT WAS MENTIONED BUT NOT DISCUSSED. INDEED; MY INSTRUCTOR NEVER CHKED MY FLT PLAN FOR DENSITY ALT CALCULATIONS (I DID HAVE THEM) NOR DID WE DISCUSS HOW EXTREME HEAT CAN PLAY A ROLE IN PLT SAFETY. THE FOLLOWING ITEMS SHOULD BE COVERED; TAUGHT; AND PRACTICED (WHERE POSSIBLE) AS PART OF A TRAINING CURRICULUM: DOOR OPEN: WHAT IT SOUNDS LIKE; FEELS LIKE; HOW THE PLANE HANDLES; WHAT TO DO IF THE DOOR SHOULD OPEN IN FLT. ABORTED TKOF: PRACTICE WITH INSTRUCTOR; KNOW HOW THE SUDDEN LOSS OF PWR WILL AFFECT THE PLANE AND THE WAY IT HANDLES HOW TO BRAKE EFFECTIVELY; PWR TO IDLE OR MIXTURE OFF. EMER EVAC OF ACFT: WHAT TO TURN OFF; HOW TO TURN FUEL OFF; WHAT TO TAKE (IF POSSIBLE); OPEN DOOR BEFORE LNDG. HEAT AND DENSITY ALT ISSUES: LONGER ROLL ON TKOF AND LNDG; HEAT INSIDE THE AIRPLANE; HOW TO PLAN FLTS AROUND THESE ISSUES. I BELIEVE MY INSTRUCTORS ARE GOOD TEACHERS OF THE PRESCRIBED CURRICULUM; BUT THEY DO NOT ADD TO AND EXPAND ON IT.