I removed the bug icon (right-click, Panel options -> remove this panel) and the entire taskbar disappeared. I was able to get it back by right-clicking on the desktop and choosing to add a default panel. Why did this happen?
Because you chose “remove this panel”. What else would you expect to happen than the whole panel to disappear?
The updater applet is a part of the system tray. If you want to remove just that, right-click on the system tray (the easiest spot would be the small up arrow on the right end) and choose “System Tray Settings”. You can disable all system tray applets there.