What is this grey bar underneath the home icon row?
Oddly enough I have been having an extremely similar problem. There was a small bar that showed up under the Dock on my desktop, about as long as the Dock is wide (my Dock is vertical) and with rounded corners. It was black, and the running application dots on my Dock are black as well (whereas yours is grey, and the bar is grey).

Upgrading from 10.10.1 to 10.10.2 caused it to disappear. Then it reappeared, along with a second one. It survived reboots. I didn't think it was actually part of the Dock, because it stayed in the same place when I moved the Dock from the right side of the screen to the bottom. Although, when the second one showed up, and I ran my cursor over the dock (triggering magnification of the icons), it stayed above the icons on the app side of the dock, and was below the icons on the Trash side of the dock.

I finally ran this terminal command to kill the Dock, and when it restarted itself, the bars were gone. It's been about a day and they haven't reappeared. Fingers crossed. I'm assuming this is a bug in Yosemite.

killall -KILL Dock