Dock Icons will not move in Yosemite
Did you try this?

Finder and Dock issues in OSX 10.10.1

I had the same exact problem and it was definitely caused by Pushbullet extension in Chrome. Disabled completely the extension, restarted the system and the dock started working. 
Don't know if it was caused specifically by the "Universal copy &amp; paste" option being enabled or the extension as a whole, would have to investigate this further. Hope this helps.