1. () is basically closer to "to make a mistake" rather than "to do something in a clumsy way". It tends to refer to funny and/or affordable mistakes typically made by a dojikko character. There is another godan verb , which may be a little more common (, , ...) and has almost the same meaning. (You cannot directly modify with , though)
2. I think this answers your question, although there is not even a known kanji version in this case. Few people know or care about the etymology of .