I found that this is a specific problem with the way that ObjectDock deals with the system tray icons. I found that if I unhide the Windows Taskbar at the bottom and use the system tray there, the problem does not occur. So, in order to work around the problem, don't use the system tray icons in ObjectDock itself, use the taskbar for that.
I'm not fond of the solution myself, since I like to keep the taskbar hidden, and I've opened a ticket with Stardock (TKH-716546). Here is a transcript of my ticket:
I am opening this ticket because I am having a problem with ObjectDock. I am running v1.90 535u, and Skype v220.127.116.11 on Windows XP SP3.
I am running four docks:
1.) Non-tabbed dock with a 'Start Menu helper'.
2.) Tabbed dock with application icons.
3.) Non-tabbed dock with running applications.
4.) Non-tabbed dock with tray icons.
Also of note, I have the Windows Taskbar minimized.
Here is an outline of the problem, and the steps to reproduce it.
1.) Start ObjectDock.
2.) Start Skype.
3.) Minimize Skype to the taskbar by pressing the "X" button in the upper right hand corner.
4.) Un-minimize Skype by double-clicking the Tray Icon (in the #4 dock).
Now, Skype will not respond to any mouse or keyboard input (menus, chat, 'X' button), and has to be terminated from the Task Manager in order to be shut down. This condition persists even after ObjectDock is closed.
The problem does not exist when Skype is launched and used when ObjectDock is not running. Nor does the problem occur if I un-hide the Windows Taskbar and use the system tray icon there. The problem directly stems from using the tray icon in the dock specifically containing my system tray icons.
Now that I think about it, the same thing happens when I run the Realtek HD Sound Effect Manager for my sound card on my PC.
There is a forum post on wincustomize.com here: https://forums.wincustomize.com/338912
There is a bug report to Skype here: https://developer.skype.com/jira/browse/SCW-1008
I agree with the Skype developers that this really isn't their problem and is something you should address, since it also seems to happen with other applications. Please help me (and the other users of ObjectDock) find a solution to this problem other than the workaround that I have provided.