1) In theory android shouldn't have a need to close apps; they just sit there unless another program needs memory, in which case the least recently closed app is shut down. To be honest though I do use a task killer myself.
3) The only other thing I can think of is that, when you read the texts, do you read them as popups? You know, where it apears over whatever you're doing rather than going to the app. It can be set so that reading via the popup doesn't mark it as read.