Is this a real Mega board or a clone? The reference design has other circuitry associated with pin 13 to power the on-board LED, which could be interfering with the sensor. If it is some clone, then there is no guarantee they followed the reference design and may have eliminated/changed things.
Yet another lesson in why Pin 13 should not be for first choice for use...
As for the standard chess notation, all those strings "A1", "A2", etc will take up a ton of memory. It would be much better to use uint8_t and have row 1 be 0..7, row 2 be 10..17, etc. and then when you go to actually print them out, convert them to the standard notation.