Ethernet Shield on Arduino Mini
You can just connect the pins used on the shield the the corresponding pins on your custom board. Check the Schematic to see what pins are used for what (and which can be left unconnected)

The only thing you need to lookout for is, that the ethernets shield seems to require 3.3v (the shield has it's own 3.3v regulator, so you only need to connect 5v)

That 16u2 doesn't do anything but usb-to-serial. It's not needed for the ethernet shield.

The ISP header is just 5v, GND, and the 3 SPI pins and reset. But those SPI pins are also broken out to the pins 11, 12 and 13.