Moving Icon location on Taskbar & Resizing

That's normal. Plenty of programs run in the notification area (which is often times erroneously called the system tray). Unless the main window is open, you will not see a large taskbar button for these programs. Even assuming you pinned such a program to the taskbar while the main window was open, as soon as you closed that window the taskbar button would revert to an "unlaunched" state. There's absolutely nothing to do about this except check to see if there is an updated version of the program. Of course I do seem to recall weatherbug being full of bugs. The kind that spy on your behavior and report back to their corporate overlords.

Have you looked at the weather app on the Start screen that comes with Windows 8? It does pretty much everything I remember weatherbug claiming to do and you could pin it to the task bar if you want. Plus it likely isn't spying on you... At least not to the extent of weatherbug does if memory serves.