How do I build a paper feeder with LEGO to transfer a single sheet of paper from a deck robustly?
There are many ways you can approach this problem using LEGO. I would recommend the 8547 Mindstorms NXT 2.0 set, as it includes the software required to program a basic paper feeder including a microprocessor (NXT Brick), three motors, four sensors (ultrasonic, two touch and a colour sensor) as well as plenty of other parts which you can use to construct the paper feed.

Set 8547 also comes with four wheels with rubber tyres which can be powered by the motors to feed paper. These tyres will have more enough traction to pull without damaging the paper.

A colour sensor can be used at a fixed point to detect the start of a new page or a motor can be calibrated to perform an X amount of rotations required to feed one sheet of paper. However if you wish to feed the paper, you need to consider the strengths and weaknesses of the method used as well as when and where it is likely to fail.


  I am new to LEGO but I think creating the apparatus with LEGO can reduce the cost.


You may be able to reduce the costs by purchasing the essential parts from Bricklink. Although it can't be guaranteed that you'll be able to get all the parts in one purchase and it may be more expensive to buy the parts separately. 


  I will use for processing more than 2 million pieces of paper.


If you're looking for something reliable that can scan that many sheets of paper, you may find it more economical to have find a professional archival company that can process your documents/images.