Suggestions about Mega-equivalent with built-in Ethernet

shouldn't that be 7-20V?

No, that's deliberate. The combo of the pair of MCUs and the Wiznet Ethernet chipset draws a significant amount of current, and supplying anything more than about 12V to the onboard linear regulator can push it closer to the thermal limit than I like: particularly if you stack another shield on top that draws lots of current, such as an LCD shield with a backlight. A combination of boards like an Uno with an Arduino Ethernet Shield and an LCD shield has the same problem. If you feed it 10V it'll be quite happy. At 12V the reg gets very warm, and at 20V it would almost certainly go into thermal shutdown.

You can feed it more, but the 7-10V range indicated on the overlay is a recommendation that gives a bit of a safety margin.

Jon