How can I safely connect both Arduino and a 120V 10A circuit?
As Kevin commented, you can connect them to an Arduino directly providing they're using the same logic voltage. There's no harm in connecting the Arduino directly to the relay control, as the control will be isolated from the rest of the relay. Theoretically, you could pass a high enough current to cause the gas inside the relay to ionise and cause electrical arcs, but at that point I'd be worried for your safety, not the Arduino's. Basically, as long as you're careful when dealing with high currents, then you and your Arduino will be fine.

EDIT: In response to question 2, I doubt you could use just one power cord - you would need one connected to a 120V, 5A (600W) source, and one to a low current one. The exception to this would be if you used a modified ATX Power Supply - this would supply 600W (and so could be transformed to the correct voltage), and also has 5V logic lines which could power an Arduino - this has been done successfully in the RepRap 3D Printer.