Powering Arduino with 110v and controlling relay
In these cases where power must come from mains, my preferred solution is to buy a cheap phone charger+usb cable from ebay. They are sealed and safer than do-it-yourself solutions. The power adapter is barely larger than a normal plug for mains.

If you absolutely need to reach the arduino with only one cable, you can still use the solution i gave, only use a power cord to reach the contraption, then use a socket expander for mains and very short usb cable.