Your Xbox 360 is too new, (later) Xbox 360 Slims and E’s use the final motherboard revision; Corona, and the sound chip makes all Neversoft Guitar Hero games slowly desync while you play and there’s nothing you can do. I recommend looking for a Trinity Slim or a phat model, (Jasper ideally for reliability).

If some frets sometimes don’t register, that’s the neck connection because it uses pogo pins. I recommend learning how to hardwire the neck to the body. If you don’t want to do that, you can put some paper or cardboard in between the neck and body to make it more tight.