Can Only Upload Arduino Sketch Once to Custom Mega2560 Board
Your RESET circuit is wrong.

You have no pullup resistor on the RESET pin of the ATMega3560.
You have an extraneous pulldown resistor on the D7 pin of the 16U2.

That means your 16U2 will not be able to trigger a reset of the ATMega2560 to run the bootloader.
Remove the pulldown on D7 of the 16U2 and add a 10kΩ pullup to the RESET pin of the ATMega2560. Without that pullup:

The chip may not run properly as RESET is floating (it may have an internal pullup, I can't recall)
The capacitor linking the 16U2 D7 and RESET of the ATMega2560 won't be able to generate a proper reset pulse. This will be exacerbated by the 1kΩ pulldown.

On the Mega2560 reference schematic RN1A is the pullup in question.