It's normal for all the apps to go grey on top. You should be looking at the bottom of the screen for running or recently used apps.
"phone" icon? What does this mean, are they using battery? Am I not closing them properly, so they won't show up there?
The majority of apps showing on your multitask dock should just be in a frozen/saved state, until your open it again. So it shouldn't waste battery. There are some apps that will stay running in the background though such as gps apps, pandora, skype, etc.
Apple says multitask doesn't waste battery, but many iPhone owners say it does. Here are some articles:
Steve Jobs: "Trust the iPhone", Multitasking Don't waste battery life!
Does iPhone iOS multitasking drain my battery life? | The iPhone FAQ
Honestly I don't know for sure, but I sorta think it does. I'm always completely closing apps throughout the day. Just my opinion though...