How can I look inside iOS apps' folders from within iTunes?

Unfortunately there is no way. The list is simply a container with a list of files/folders that can be sent between the two devices. The files themselves are stored on the source, and only actually moved once you save them, or sync, depending on the direction.

In short, it's a plist saved with a simple link to the files/folders, which is then retrieved when you perform an action. As far as I know, not even files can be opened directly from there without downloading them first.

In all seriousness, have you considered Dropbox as a medium for this?

P.S.: Double-clicking should allow you to rename, or it does on my version of iTunes.

You cannot using iTunes, however products such as PhoneView allow you to browse inside folders under the App and transfer files also, all outside of iTunes.