So, when you drag an app icon off the Dock, it does not go "poof" and disappear?
If so, first, you may want to try the good ol' Repair Disk Permissions, in Disk Utility First Aid tab, on your startup disk. Don't worry if a lot of messages are displayed. Things that need to be fixed are being fixed, and the rest of the info is mostly "informational."
If that does not help, my only other idea is to get rid of the current Dock user preference file. This will "reset" the Dock an put it back to its initial default state. So any app icons you added/removed, and changes you made to the Dock's appearance, will be gone. It will look like it did when you first used your user account. If you want to do this...
I just tried this in my "Test User" user account, to confirm. First, you need to go into your user account Library folder. This folder is hidden starting with Lion, so from Finder's Menu Bar, select
Go -> Go to Folder...
In the field, enter
/Users/[your short user name]/Library
where [your short user name] is the name of your user home folder. And click Go. This will open a Finder window showing the contents of the hidden Library folder. Open the sub-folder named Preferences. Find the file named
and drag it to the Desktop. Log out and log back in. The system should create a fresh Dock user preference file with the Dock in the initial default state. Hopefully, it won't have the same problems and you can update the Dock as desired.