Micro controller to set 'dial knob' mode on multimeter
Can a microcontoller be used to replace the "dial knob" function in a
  'off the shelf' multimeter(10A max AC/DC).


On some ranges perhaps, depending on how the meter uses the switch contacts. Some contacts may be used to route analog signals that cannot easily be switched with a transistor or analog transmission gate. You need to determine what every contact does, which will be tricky unless you have a complete circuit diagram. If you are not careful the meter's accuracy could be compromised - or worse, something might blow up (if it can't handle the voltage and current being switched). So you may need a combination of transistors, gates, optocouplers and/or relays to get full functionality - quite a mess!     

I would do it the easy way - mechanically. Attach a small geared stepper motor to each 'dial knob', then you can rotate the knob under MCU control. You may also need a 'home' switch to put the knob in a known position at start up. One of the existing switch contacts may have a signal that can be used for this purpose (eg. power on switch). 


  How far can the LCD be lifted off the PCB?


Most meters use a conductive rubber strip to connect the LCD to the board. To extend this you would have to provide another circuit board to clamp the LCD screen and strip onto, then join this to the meter PCB with a ribbon cable. LCD drive signals are relatively low frequency and very low power, so the cable could probably be made quite long.