Will leaving multiple apps open on iPad 3 drain battery life?

It depends upon what the apps are. A lot of apps (especially older ones) aren't coded to make use of multi-tasking, so effectively they just become 'recently used' when you switch into a different app and they don't remain active. There is no harm in closing them via the taskbar.

I just got into the habit of closing off everything from the taskbar after I'm done with it. If you keep it managed (closing an app right away after you're done with it), it shouldn't take but a few seconds of extra time closing the few apps you forgot to close.

The answer is no. Unless the background App is doing something like streaming music then it isn't running at all.

Most apps will go into a dormant state when closed and use no CPU. Those that do continue to run in the background are often obvious, e.g. you will continue to hear the sound from a media player. From my experience, the ones to look out for are the ones that use GPS. If you happen to leave one of those running (in the foreground) and simply have the screen turned off, they will eat your battery.