Live stream from GoPro to YouTube or another video service
No GoPro is appropriate for the job by itself.  Any of them (or even a basic webcam) could work as a camera, your problem is that you need something that can encode and send a stream by itself.  That is a specialized device and not something a GoPro can do.  You would want a either a device built for doing a direct stream (such as an IP camera) or you would want to have a camera with a clean HDMI (or other) output that can be fed in to a stand alone streaming device.

For the second half, recording quality and stream quality are completely and 100% independent of each other.  They are both two different captures of the same visual information, but the sensor is going to pick up maximum quality regardless of if you are recording in some low bandwidth format or not and the steam is going to run off the output which should be the high quality format regardless of what quality you are streaming at.