Detecting chess pieces on the board
Read range on RFID tags is determined by the antennas on both the reader and the tag. You often don't have much control on the tag, but you can control the sensitivity of the readers antenna. You could use an RF multiplexer and a 125 kHz reader like the ID-3LA and sequentially read each location on the board. Not a trivial task, but it can be done.
One thing that would be interesting is to make the chess board from a PCB with copper trace reader antennas, the efficiency may be low, but that may be what you want. If your range was &lt;2 mm, it shouldn't be a problem. Have the PCB made with black solder resist and silk screen the white squares. All the components would mounted on the other side of the board making the top very nearly flat.
As an added reference, Texas Instruments Application Report SLOA167 discusses a 4x4 array of RFID antennas.