In battery stats I have the option to STOP certain apps. I can't quite see the purpose of this. If I do stop them, and check later, they've started again. Is it meant as a battery saving measure before sleep?
No, it's really more of a troubleshooting feature. If an app is misbehaving, you can stop it. It's the same as going to Settings>Apps, selecting the app, and tapping Force Stop.