Finder and Dock issues in OSX 10.10.1
Solution 1:
I know it may seem totally unrelated, but if you have the Pushbullet extension installed for chrome, try disabling "Universal copy & paste". You may need a restart afterwards.
Solution 2:
Repair Permissions cannot possibly fix this problem.
Repair Permissions only sets permissions on files/folders back to the permissions they were installed with, and only if they were installed using Apple's Installer. (And it doesn't even do that well.)
NOTHING in your home folder was installed that way.
NOTHING in your home folder will even be looked at by Repair Permissions.
Not your Dock. Not anything on your Dock. Not Dock's preferences.
Let's look at things that might actually cause the symptoms described.
First, there used to be a problem where people would accidentally drag things off their Finder sidebar and wonder where they'd gone. Apple fixed that by requiring you to hold down the command key while dragging items off the sidebar, which was consistent with the way toolbar items and menubar items work. That led to a rash of queries: why can't I drag anything off my sidebar any more? Yosemite went back to the old behavior: you no longer need to command-drag to get things off the sidebar. (Instead, it seems you now need to command-drag to put something on the sidebar! I wonder whose bright idea that was?)
Why am I talking about the sidebar when the question is about the Dock? Because we expect the UI to behave the same everywhere. You still need to command-drag to take something off the menubar or toolbar. You used to need command-drag to remove something from the sidebar. It feels intuitively obvious that you should need to command-drag to get something off the Dock.
But surprise! On Yosemite, command-drag to or from the Dock "locks" the Dock for the duration of the drag. The reason is to make it easier to drag something onto an app or into a folder that's on the Dock. By command-dragging, you stop the Dock icons from dancing away from your mouse. It even lets you drag a document that's on the Dock onto an app that's on the Dock. Normally, drag-and-drop from the Dock would remove the document from the Dock no matter where you tried to drop it. But now, command-drag from the Dock signals that you're trying to drag-and-drop the item, not remove it.
So, just as a wild guess, is it possible that you can't drag from the Dock because you're holding down the command key? Muscle memory is insidious. It makes you do things without even realizing you're doing them.
As another wild guess... I tried to use a friend's Mac, and kept running up against walls like this where things just weren't working as expected. It turns out my friend likes Sticky Keys, and has it turned on all the time. I'd carelessly press the control key to do something, then change my mind, blissfully unaware that the control key was now stuck down. Everywhere I clicked, a contextual menu would pop up surprisingly. I didn't realize what was going on because I rarely use a MacBook Pro and thought it was just my muscle memory not adapting to its tiny (by comparison) trackpad.
So, is it possible that you're accidentally turning on Sticky Keys? It's easy to do. Just tap the shift key five times without pressing an ordinary key. Then it's easy to start doing command drags without actually holding down the command key. Tap the shift key five more times to turn it off.