There's a folder shortcut on my dock, next to the trash. Why is there and how do I remove it?

Simply click on it to hold it then drag it off of the dock and release the click. It should disappear in an animated puff of smoke.

Open ~/Library/Preferences/com.apple.dock.plist in Xcode (I see you have it in your screenshot) and remove the entry for the folder (Probably Root > persistent-others > Item 0). Then, in Terminal, run killall Dock.

I had a similar issue but after realizing that the folder which couldn't be deleted was basically a "mounted network drive" like you originally mentioned I looked at Mac's solution for remote working when not on your work's network. It's called "mobile accounts". So basically I had set my Mac up to copy and sync with a particular drive on our work network so when I was away I had access to my files. You can check if this is your case by working backwards from setting up such an account (http://support.apple.com/kb/PH11307). That folder couldn't be deleted from the doc until I turned off this mobile account and then the folder was absent after restarting my machine.