How do I install the firmware on an Arduino Mega clone? I have a brand new board which is a clone of the Mega2560.  I need to install the USB firmware on the 16U2 and the bootloader on the 2650 chip.  I have Atmel Studio and a JTAGICE-3. Using those, I can download HEX files compiled with the Arduino  IDE, and they run successfully. 

But I'm having trouble finding Mega-specific instructions for getting the firmware installed so that I can use the Arduino ISP protocol for downloading new sketches. I can find much more information about the Uno than the Mega. And much of the information I do find seems to be dated.

1) Which are the right files to download?
   I've tried to install the 'MEGA-dfu_and_usbserial_combined.hex' to the 16U2 and the 'stk500boot_v2_mega2560.hex' to the 2560. 

But the IDE won't connect via AVR ISP, and I get no serial communications from a serial sample sketch downloaded with the ICE.

2) What are the correct Fuse settings after I flash those files?

3) Is there a good way to confirm that the 16U2  USB converter is working correctly in isolation from the 2560 chip?
Let focus on one thing at a time, that is flashing the 16u2. You can see if you've successfully done that when you can see the Arduino in your device-manager.

First of, MEGA-dfu_and_usbserial_combined.hex is, as stated on the github page, for the 8u2, not the 16u2. Since you already have a ISP programmer you don't need the dfu functionality.  You can just flash the precompiled Arduino-usbserial-atmega16u2-Mega2560-Rev3.hex file.

The fuses can also be found in the readme.txt on github: -U lfuse:w:0xFF:m -U hfuse:w:0xD9:m -U efuse:w:0xF4:m -U lock:w:0x0F:m. I don't see any difference in the fuses between the 8u2 and 16u2, so they should work for the 16u2 too.

Hope this helps.