PC12 pilot reported multiple instances of heading and altitude deviations possibly due to avionics anomalies.

Narrative

While climbing out of ZZZ and being vectored by ZZZ departure I received several heading; altitude and terrain alerts. My altitude in the PDC Clearance was for 9000' expect FL200 10 min after departure. When given line up and wait on [Runway] XX; tower advised wind was 260/17G22 with Moderate Precipitation over the airport extending 4 miles west. I was then told to turn left to 260 and cleared for takeoff. After takeoff and turning to 260; tower ask to verify heading as 260. At the time I thought that was a little odd. During the climb I went IMC and was picking up turbulence and light icing. I turned-on Prop Deice and boots. I engaged the autopilot around 52-5300 ft. and was turned over to departure. I made my call to departure as 5200' climbing to 9000'. They advised of precip and turbulence for the next 40 miles. I apparently missed a climb to 20000' call as I was distracted by the icing; turbulence and passengers. Going thru about 8200 ft. I was given a left turn to 190. I noticed the autopilot wasn't grabbing the preselected 9000 ft. so I disengaged the autopilot to level off and continued turning to 190. I was then given a 170 heading and I remember starting the turn and from all indications; all was in order until departure kept telling me to keep turning left to 150 and climb. Controller then started giving me altitude and terrain alerts and to continue my left turn and climb. I was starting to get confused because Attitude (bank and pitch) all appeared normal and agreed with the right side except at one point heading showed 210 and verified with controller which further confused me because I had assumed I was already heading 190. I considered switching the left AHRS and ADC to #2 as a precaution for false heading and/or airspeed indications but with the turbulence; icing and urgency from the controller never had a chance. Departure finally verified us on a 150 heading and climbing and I remember re-engaging the autopilot with 20000 ft. set in the preselect. Subsequent climb and Nav capture went normal until about :30 into the flight; and level at 220; when both airspeeds on left (Pilot Side) as well as the Standby Attitude Indicator went to 215 KTS and and gave an overspeed warning. Right side was normal at 172 KTS. I verified that Probes switch was on; and CAWs just showed the green Probes De Ice light on. The only indication on a Series 9 PC12 I would have gotten was an orange Static caution which I didn't. I did check the CB's and all were in. OAT was -19 but at that point I was not in any visible moisture. Upon descent both airspeeds returned to normal while descending. Not sure if that was related to the climb out issues but thought that was odd. Landing and return flight were normal. After having time to review the flight; I have several observations and thoughts: 1- After I was on a heading of about 190; the Foreflight Track Log showed a right turn; which would explain why I was on a heading of 210. I don't remember if I was hand flying or on AP at that point. 2-My Foreflight Track Log also shows that I had a left bank with no turn initially at the point where I thought I was turning left to 170 then 150 which was causing me confusion on my heading. 3- I have since observed on this particular airplane that when the autopilot or Yaw Damper is turned off; the rudder kicks to the right; requiring left rudder re-trim. This may explain why it wasn't turning in the left bank (#2 above). The Yaw Damper light on the Autopilot was out. 4- I did consider switching the AHRS and ADC's; thinking I was having a heading issue; but on the Garmin TXi's it involves several touch screen settings and isn't that intuitive especially in turbulence. 5- At no point prior to the controller giving me altitude alerts due to terrain; did he ask me why I wasn't climbing out of 9000 ft. I possibly could have started climbing and avoiding the altitude alerts had he ask and clarified my assigned altitude as 20000 ft. earlier. 6- I neverreceived any Terrain Alerts on either the avionics nor on Foreflight on my iPad. 7- At ZZZ they recently changed the initial altitude for departures from 10000 ft. to 9000 ft. The 10000 ft. would have offered more safety in the event such as this; with heading deviations or lost comm; especially near the higher terrain west of ZZZ. 8- I feel training and familiarization has a role to play. There are 3 different types of PC12's (series 9; series 10 and the NG) with 3 different Avionic set ups (old Honeywell DUI's with Garmin GTN 430/503's; newer Garmin TXi's with GTS 650/750s; and NGs with the Honeywell Apex). All can have different attitude displays and autopilots and there isn't much standardization even within the same avionics' packages. The plane I was flying was and older legacy PC12/45 with the newer Garmin TXi's; GTS 650/750 and the older Honeywell autopilot. Annual recurrent simulator training is required by insurance; and to my knowledge; there isn't a simulator offered that has the new Garmin avionics. My most recent recurrent was in the NG but have tried to alternate between the NG and the older legacy but they only offer one with the older Avionics.