You said at the beginning that it does not appear in iTunes, but if over the course of trouble-shooting, you got it recognized by iTunes, you can try doing a Restore on the shuffle. This will erase the iPod, re-install its software, and set it to default setting. The Restore button is on the shuffle's Summary screen in iTunes. This will rule out some type of software or settings issue on the shuffle.
If you did not have that USB power adapter, I'd also suggest ruling out a problem with the USB port not providing sufficient power. But since you still get the same flashing red light with the power adapter, it is less likely to be a problem on the USB port. However, if you are connecting the shuffle through a USB hub, make sure you try it on a direct USB port with other USB devices disconnected.