I will cover your points in order:
I wouldn't know what software is running in the firmware, even if I did I would not be able to disseminate that information. If it is running GPL code that code should be available on http://support.dlink.com/gpl. Busybox is GPL, thus I infer that this router does not run Busybox.
I know where you are going with this and I suspect it's a dead end, but I always invite people to have fun trying. Someone who actually works closely with that product would know a lot more about the fruitfulness of such a path.
You aren't finding detailed specs (chipset, etc...) because D-Link does not usually disseminate such information. I could not provide it to you even if I knew (which again I don't, I work with enterprise equipment hence the avatar)
This device does not support WDS and should not work with lazy WDS (what you got working if you were able to WDS to it), I would find the fact that it worked at all for you lucky.