Can apps be controlled through adb? Can an installed application be completely controlled via adb? 

For example:
Consider I need to to export my Contacts to SD Card. So here there are basically 5 Steps involved

1)Open Contacts application

2)Press Menu Key

3)Select Import/Export option

4)Select Export to SD card

5)Confirm Export

Can any/all of them be performed via adb?

This can be useful when you have an unrooted phone with a bricked display and all access you have is via adb.
Yes and no. You can issue intents with adb but that's it. So the steps you described can't be achieved with adb.

Edit: ce4 showed a blog post which mentions the input binary which is able to simulate key events.

adb shell input keyevent &lt;keycode&gt;