When you double click the home button on iPhone?

A row of previously used apps appear, are they running the the background and using battery life? Or is it just simply showing your latest used apps?

Depends on the app. Usually, most of them close after a few minutes of use, but they will still show down there until you hold the app and remove them from the bottom bar

It just shows you what you used previously

It does mean it's opened.
Just tap and hold and click on the "x" to make it turn off. ( like a game )

& If you do exit the message or something, it wouldn't stop texts from going or coming to you.