(a) Sorry, no. Registration does not affect your rights/ability to own or shoot a gun UNLESS you're not supposed to have one in the first place, just like backgrounds checks do now. Unless you're advocating for an/all people to be able to own a gun, you are incorrect.
Also, assuming you (yes, YOU) are legally allowed to purchase a gun, registration would make 0 difference in your ability to own/fire a gun.
(b) Seeing as there is no registration for firearms in the US, I fail to see how you can reach that conclusion. Secondly, registering of cars/licensing of drivers works pretty damn well. Get pulled over without a license, go to jail. Get pulled over in a car that's not registered to you (i.e. stolen), go to jail.