1. You have unlimited data, so even if you were signed on and it were using data constantly it wouldn't give you a crazy large bill.
2. AIM doesn't run in the background, the reason you are still signed on is because AIM didn't tell the AOL servers to set your account as "signed off" when you quit the application. So it is not running in the background, and it is not using data, even though you are technically signed on.
3. Instant messages are not SMS messages. You can send an instant message to a phone number, this still does not charge you for a text message, thought it may charge the recipient. Nothing YOU do from the AIM app will make SMS charges show up on YOUR bill unless you instant message your own cell phone number which would be pointless.
4. To set AIM to sign you off when you close AIM, go to your iPhone Settings app, scroll to the bottom and you will see AIM, change "sign off on exit."
There you go.