Hide System Tray Icons on OS X – Step-by-Step Instructions

kdocker ... it also does not appear to be able to iconify a window when closed (it actually closed instead).

As far as I understand no docker can do that. This has to be supported by the application (e.g. claws, vlc, etc). When you click on close the window manager signals the application to close, depending on its setup it closes or minimises to tray.

What I want to iconify is xfce4-terminal, so any terminal which can iconify itself and works with xfce would be ok as a solution, too.

I don't have an exact solution to this, but in terms of how it appears in window or Alt-Tab lists its identical. You could use a drop down terminal like guake. It lets you define a hotkey that will make the terminal appear and disappear. You can also add some window events as triggers for it to disappear (e.g. some other window raises itself). You can also configure other how and where it appears, what size it has. I like the fact that the terminal is always at the same place no matter when I call it and its always running at login.

The only missing feature I have found is, I can't control which display it appears on for dual display setups; it always uses the left most display.

I figured out that the application responsible for these two icons is a background service called "VAIO Event Service", which is started with Windows. It's generally not a good idea to stop this service, because in doing so, various shortcut keys stop working. The service description also mentions that power management is disabled.

Simply put: the icons cannot be disabled without losing some functionality of your VAIO notebook.

It may nevertheless be insightful to share how I figured out which app was responsible for the icons:

1. Firstly, to get some more information about the application responsible for the tray icons, I expanded the notification area and clicked "Customize..." to open the Windows Notification Area Icons window. From there, I determined that the application name is "VAIO Control Center".
2. I then used Sysinternal's Process Explorer to review all running tasks that contain that application in the description column. Sure enough, there was a match. In the mouse-over tooltip, the match revealed that it was associated with a service.
3. In the Windows Services window, I located that service and stopped it. The icons immediately disappeared from my tray, but I could no longer disable the touchpad with the Fn+F1 shortcut.