What you want is to make it more difficult for people to own a firearm. Which is where the phrase 'shall not infringe' comes into play.
Wrong. Infringement means the law denies you the right. Such as D.C.'s law (addressed in Heller) effectively banning handgun ownership that stripped ordinary citizens of their right to defend their homes. Reasonable regulation - even regulation that may inconvenience you - is perfectly legal, as has been found in case after case, including Heller itself.
Registration does not make it more difficult for you to own a firearm, it is merely a device for identifying that you are the person who has legal responsibility for it. Does registration deny you the ability to own a car? Of course not. The argument is absurd.