When I minimize a music player it just makes my already cluttered taskbar worse and closing it just makes whatever I was playing stop. So, is there a setting to minimize a window to tray (along with the system tray) so it won't get in my way? If not, then, please post a workaround or something that could help ...

Which music player are you talking about? This always worked fine in Amarok (closing the window by clicking on the close button doesn't exit the application), and should also work in Clementine which is an Amarok fork.