You defiantly want to get the Hero 7 Black. It's internal Image stabilization is awesome and video quality is great. As long as your not bouncing over boulders at high speed you will get a pretty stable video. They make various mounts for different diameter bars.
Yes, you can use the Gopro with it's usb type charging cable plugged into a cigarette lighter adapter of the correct usb type as your cable. With the Hero 7 black 12v to 5v works, and I believe I read somewhere at 12v a little over 1amp was working. Apparently it won't charge the battery while recording so it will use the gopro battery birst down to about 3% then use external power or you can just use external power with the gopro battery removed. There are some issues people have had with freezing while connected to external power, some have solved this by turning wifi and other features they aren't using off, others have cut the end of the charging cable off and only used the + and - wires to hard wire it to the vehicle power source and it removed those issues also.