I have a question, when mic recordings of the girl's voices appear on the internet, does that mean the perf was lipsynched?
There's no corellation between the existance of mic recordings and the performance being live or not.
All the girls wear radio mics and they're all on different frequencies all being fed into the desk, with a fader(volume control) for each.
It's the sound engineers job to pull up the fader on the girls voices when they're singing. If they sound like crap (like some of them do, because they're out of breath from dancing) then the engineer will pull the fader down low (or off completely) and you'll just hear the prerecorded vocals. Sometimes a performance is all lipsynced, sometimes it's all live, sometimes it's a mixture of both.
Regardless of whether the fader is up or down, a wota can still tune into the radio frequency and record EVERYTHING the mic picks up.
I wonder if the girls ever sometimes say things into the mic that aren't so savory..
They do. There's that recording from Nacchi's grad concert where Rika was saying the fans were silly for making cat noises during an audience participation bit. There's been others, and probably many others we never hear about.