After all the elapsed time since my first posting I am very disappointed that Captain Sim hasn't responded.

There are only two possible reasons for this.  They can't or they won't.  

I won't be buying any more of their products.

Thank you for your response.

I looked through the Technical Support Section and the only fix I found was one which addressed another problem.

However, I redownloaded the aircraft and it now seems to work correctly.

If you want to get Autopilot to work its very easy...

Fist, take-off manually.
Second, climb to a good altitude (most times 10,000 ft)
Open the Autopilot window.
Turn up the Autopilot Aileron and Elevator Channels Master Switches (those 2 little things on the right hand side of the Autopilot Turn and Pitch Controller (circle thing))
If you want to keep that altitude then turn on the Altitude Hold Switch (on the right of the of the Autopilot Aileron and Elevator Channels Master Switches)
Now, the Autopilot will hold this altitude.
You can use the Autopilot Turn and Pitch Controller (the circle thing near the Autopilot Aileron and Elevator Channels Master Switches)
to turn your aircraft.
Pull the button right to make it go right, left to make it go left...
and if you want it to go up, disengage the Altitude Hold Switch and pull to go up and down to... well, go down.
If you want the airplane to go straight as well, then turn the heading bug (near the Horizontal Situation) to where you want to go and turn on Heading Select Switch (far right button in the window)

And that's how you turn on Autopilot. If you want to turn off autopilot then...

Turn off the Altitude Hold Switch
Turn off the Heading Select Switch
Disengage the Autopilot Aileron and Elevator Channels Master Switches
You will hear a warning bell. Don't worry, you haven't done anything wrong, its just to tell you that autopilot is now off.
Push the Autopilot disengage warning light (should be flashing red)
and that's it! I always use autopilot when flying for long distances. But, YOU should always be landing and taking-off with autopilot disengaged.