When you pull down the notification bar you see a list of recently used apps, but they're not necessarily running.
Bluetooth will only be on if you have a little icon at the top near the wifi / 3G battery icons. If this icon isn't there then it's off (check in settings if you must).
To close all apps hold the power button until you get the power menu and then select "Restart". This will close all apps and restart the phone. You shouldn't really have to do this very often though.
To answer your other question - To remove desktop shortcuts, press and hold the icon until it lets you move it around and then drag in down to the launcher dock where you will see a trash can icon. This doesn't uninstall the app / shortcut, only removes it from that screen