That is strange.
Actually that was one of my first questions when I got the S4, how to see the preview of the text message when it comes in. I don't think it was on by default and just found it from messing with the app settings.
Yeah not sure on this one. I was in the opposite boat as you, it was doing it to me and I couldn't turn it off. I am still using GoSMS and like it better. That would be my only suggestion, they have a way to pop it up when you get a new message. Plus many themes to make it look a little nicer.