This is normal behaviour. even when you turn off your PC, the lights on the graphic card and the motherboard stays on, indicating your power has been plugged correctly. at least that's how it is on Asus hardware. There is no thing to worry about.
That is strange, what lights are on? I have not used an Asus board so I am not sure if that is normal, I know Gigabyte and MSI have no lights on when off. Are you sure your computer isn't in some sleep state?