Where are the items in the system tray stored?

Are they in the registry or in some folder?

Generally to remove or configure these tray icons you'll need to open the application itself and configure the "show in system tray" option if there is one.

Applications decide whether to put an icon in the System Tray, what it will look like and its features when it gets an event (mouse click).

The System Tray is just a "container" window for icons (to make it short). There is nothing in the Registry nor a folder that contains or describe what the System Tray contains at a specific time.

At most, the Registry or .ini files might contain application-specific settings, some of them driving their behavior regarding their eventual system tray icon.