A DA20 FLC DEPARTING MKE FAILS TO COMPLY WITH ATC INSTRUCTIONS RESULTING IN AN ALT AND TRACK DEV.

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THIS OCCURRED ON THE 11TH LEG OF ROAD SHOW; 2ND LEG OF DAY. AFTER DEPARTING MKE; WE WERE HANDED OFF TO CHICAGO DEP (I THINK). THIS CTLR ASSIGNED US A HDG OF 90 DEGS OR 95 DEGS AND CLB TO 11000 FT WHICH MODIFIED OUR BREW 3 SID. I WAS USING AUTOPLT (COLLINS AP104) FOR CLB AND HDG. AT 10500 FT OR 10700 FT; CTLR WANTED TO KNOW OUR HDG. HE WAS ABRUPT AND BOTHERED. PNF RESPONDED WITH '090 DEGS.' CTLR INSISTED THAT WE WERE NOT TRACKING 090 DEGS AND GAVE US A TURN TO 030 DEGS. I STARTED TO COMPARE COMPASS SYS AND GPS TRACK AFTER SELECTING 060 DEGS IN MY HSI BUG. AT THIS TIME I CAUGHT US CLBING AT 11300 FT AND PUSHED THE NOSE OVER TO LEVEL AT 11000 FT. IN THE OLDER FA20'S AN ABRUPT PUSHOVER OR PULLUP WILL PRODUCE A DEFLECTION IN ALTIMETER AND VSI IN THE OPPOSITE DIRECTION UP TO 300 FT ON ALTIMETER AND 1000 FPM ON VSI. MY PUSHOVER GAVE A BRIEF CLB INDICATION ON ALTIMETER AND VSI WHICH PROMPTED ANOTHER QUERY FROM ATC REGARDING OUR ALT. AT THIS TIME I WAS LEVELING AT 11000 FT. I WAS STILL AT 060 DEGS AND WAS NOT GOING TO TURN TO 030 DEGS UNTIL I COULD ASCERTAIN WHAT MALFUNCTION WE MAY BE HAVING. THIS INCLUDED TURNING OFF WINDOW HEATS TO GET AN ACCURATE WET COMPASS READING AND CHK TRACK ON FMS. CTLR CAME BACK AND WANTED TO KNOW IF WE HAD TURNED TO 030 DEGS. THE PNF RESPONDED SHORTLY AFTER WE DETERMINED THAT OUR COMPASS SYS WAS ESSENTIALLY CORRECT AND I STARTED THE TURN TO 030 DEGS. ALTDEV WAS ESSENTIALLY RESULT OF STRESS AND CONFUSION IN COCKPIT CREATED BY CTLR'S TIMING OF INQUIRY REGARDING HDG. CTLR'S DEMEANOR WAS COMBATIVE WITH FIRST INQUIRY REGARDING OUR HDG. ALSO; AUTOPLT DOES NOT HAVE PRESELECT/LEVELOFF MODE. CTLR'S TIMING AND DEMEANOR REF TO HDG RESULTED IN PNF MISSING 200 FT BEFORE 11000 FT CALL AND MY NOT HEARING ALERTER AT SELECTED ALT AND 200 FT. A SIMPLE REQUEST FOR A TURN FURTHER L OF FIRST ASSIGNED HDG WOULD HAVE AVOIDED THE WHOLE SIT. I ALSO SHOULD HAVE RELINQUISHED EITHER TROUBLESHOOTING OR FLYING TO PNF (COPLT). THIS WAS A NEW FO AND HIS FIRST TRIP WITH OUR COMPANY. MY HESITATION IN LETTING HIM TROUBLESHOOT OR FLY OVERLOADED MY CAPABILITY TO DEAL WITH SIT.