I don't know a ton about duolingo as this is my first language, and I apologize if this has been asked before.
Is it possible to change the badge/icon for each level you gain? I think it'd be cool/feel more gratifying.
Not sure what you mean by the badge/icon? If you mean your flag next to your name, then no. Your level can change but the flag of the language you are learning cannot.
Yeah, this level: http://i.imgur.com/zJUnSg5.jpg
That's REALLY unfortunate that it can't be changed, it would give a huge feeling of progression/accomplishment. The level kind of feels empty without it changing. I imagine it having like cool border that turns from bronze to gold or diamond as you progress.
Well as you progress your tree turns gold :) and also at the end of your tree you get a golden owl.