Good questions. I haven't tested the Sonys but have had good results with the Olympus recorders. (I bought a WS-200S after writing this review and carry it daily.)
Maybe you can arrange to have your daughter sit close to the teacher, which would improve the recordings. I get noticeably better results using external mics, but that would be a hassle in daily use.
That said, an iPod with an external mic might offer the ultimate in kid-coolness. The latest iPods finally offer high-resolution recording. And the newest Olympus WS-series voice recorders play MP3s, so that's another option, although they have plastic cases and tiny buttons.
I find that stereo recordings are more intelligible than mono ones, but you'll of course need headphones or stereo speakers to get the benefit. The fingernail-size speaker on the voice recorder is low-fi.
Rechargeable batteries do work.
Check the Dragon site for voice recorder compatibility.
I have a summary of voice recorder questions and answers at http://www.oreillynet.com/digitalmedia/blog/2006/09/your_voice_recorder_questions.html.