No, it will not send live video out the USB - no camcorder does that to my knowledge. However, there may be a device out there that records video via the HDMI output. And yes it does record audio via its internal mics.
I always record to an SD card and play it back on a computer. I do not know why one could not hook up the camera directly to a computer. Some times I have had trouble running the camera on the power cord. But if the battery is charged I can get about 45 minutes of record time. Sound has been a problem. At out door events (like shows) I can get good sound recording. But in the classroom I have to have the camera within four feet of the speaker to get a good sound recording. When I plug a mic into the camera I get no sound at all.
You don't need to plug it into your computer if it is charged, but while it is charging in the wall you can record at the same time. It does record sound but if you would like the sound to be heard better then I suggest buying an external microphone.