No,apps are always running unless you shut them down..
No they aren't, at least not most of them. Not in any meaningful, battery using way. On an iOS older than 4, not at all. On iOS 4, some apps can run in the background in a very limited way (mainly GPS trackers, things that play music etc.) which can use battery, but it will be obvious if they're running (your satnav is going, or you can hear music).
As a developer, it is slightly annoying, and limits what applications can do quite a lot, but you can see why they do it – they don't want people to have to worry about manually closing applications ever, they're aiming at a demographic who don't want to have to think about multiple apps running at once.
Joe