RDU CTLR DESCRIBED OPDEV WHEN ACR DEP FAILED TO COMPLY WITH ALT RESTRICTION; ENTERING ADJACENT AIRSPACE WITHOUT HDOF.

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I WAS WORKING RDU DEP. ACFT X CHKED IN OFF OF RWY 5R RPTING LEAVING 2600 FT FOR 7000 FT. I RADAR IDENTED THE ACFT AND ISSUED A L TURN HDG 240 DEGS; RESUME THE PACK6 DEP; CLB AND MAINTAIN 12000 FT. THE ACFT READ BACK HDG 240 DEGS; RESUME THE PACK6; MAINTAIN 16000 FT. I CAUGHT THE INCORRECT ALT. ABOUT 30 SECONDS LATER THE PLT REQUESTED A HDG VERIFICATION. I INFORMED THE ACFT I WAS MORE CONCERNED WITH THE INCORRECT ALT READBACK THAN HIS HDG. THE ACFT READ BACK 12000 FT. I NOTICED THE ACFT IN A R TURN (THEY WERE INSTRUCTED TO TURN L) AND INSTRUCTED THE ACFT TO CONTINUE THE R TURN TO A HDG OF 140 DEGS AND AGAIN TO MAINTAIN 12000 FT. THE PLT ACKNOWLEDGED WITH A CORRECT READBACK. IT APPEARED THAT THE ACFT BEGAN A L TURN AND I ASKED WHAT HDG THEY WERE TURNING TO. THE PLT READS BACK 240 DEGS (WAS ASSIGNED 140 DEGS) AND SAYS WE'VE GOT IT UNDER CTL. THE PLT THEN COMES BACK AND ASKS FOR A HDG AND ALT VERIFICATION AGAIN. I INSTRUCTED THE ACFT TO CONTINUE THE R TURN TO A 230 DEG HDG. THE TWR BEEPED IN MY EAR WITH COORD AS THE ACFT READ BACK HIS HDG OF 230 DEGS AND 'MAINTAIN 16000 FT.' I MISSED THE ALT READBACK DUE TO COORD WITH THE TWR. THE ACFT HAD ALREADY ACKNOWLEDGED 12000 FT ON 2 PREVIOUS OCCASIONS AND THEN REVERTS BACK TO 16000 FT FOR SOME REASON. THE PLT ASKS ME IF HE WAS TO TURN TO AZELL INTXN AND RESUME THE DEP; AND I INFORMED THE PLT HE WAS ON AN ASSIGNED VECTOR OF 230 DEGS. I WAS BUSY WORKING SEVERAL OTHER DEPS DURING THIS TIME AND 2 OVERFLT ACFT. I OBSERVED THE ACFT HAD CLBED THROUGH HIS LAST ASSIGNED ALT AND INSTRUCTED THE ACFT TO MAINTAIN 12000 FT AND I THEN COORD WITH THE OVERLYING CTR. THE ACFT NEVER LEVELED AND CONTINUED TO CLB. THE ACFT WAS THEN ASSIGNED FL230 AND ADVISED OF A POSSIBLE PLTDEV. I LATER LEARNED THAT THE TWR HAD TO ISSUE THE INITIAL HDG TO THE SAME ACFT 2 TIMES DUE TO INCORRECT READBACKS. THE FLT CREW'S INATTENTIVENESS CAUSED UNDUE STRESS AND ADDITIONAL WORKLOAD AND COULD HAVE CAUSED A SERIOUS ACCIDENT HAD THE INCORRECT HDGS; TURNS; AND ALTS CONFLICTED WITH ANOTHER ACFT.