Also, after you press the home button, not only can you swipe out some apps to close them, or close them all, you should also see the very Task Manager icon you seek on the left hand side of the screen. Press it, and voila, your full task manager appears.
That's false and true at the same time. If you close properly coded apps with the BACK button they will not run in the background. The app will be cached and use 0 processes. If you hit the HOME button then they will always run in the background.