That chip that's getting hot is the fuse. If it gets hot and the power light goes out it means you are drawing too much current off the USB. It is made of a doped polymer that has a high positive temperature coefficient (becomes less conductive as it gets hot).
Pull the Ethernet shield off and carefully inspect both boards very carefully with light and a magnifying glass. Look for a small short, like ball or flash of solder, or even a little bit of trimmed wire. Brush both sides of each board with a medium stiff brush (old tooth brush works well) and blow them of with compressed air if you can.