Some applications seem to always stay open, even after I exit them. How do I close these? When I hold the Home icon down, several apps appear that I thought I exited - how do I actually close these?
The apps you see when long-pressing the "Home" button (or with newer devices, the "MultiTask" button with the two rectangles) are not the "currently active", but the "recently used" apps. With Android 4.0 and higher, you can simply swipe them out of that list -- which will not only remove them from the list, but also "kindly ask them" to close themselves.

For details on what happens with apps swiped out of this list, please see What actually happens when you swipe an app out of the recent apps list?