To begin, I need to move to the beginning and that would be England and King John and Magna Charta. This was an historic document for a reason seldom mentioned, this forced a King, whose rule was derived from Divine Right to acknowledge rights that preceded and surpassed his. You must take this course if you follow the logic of the proposal that King John could not act outside the Charter despite his claim to rule with God's approval, thusly if there is something that the King is prohibited there must be a countervailing authority. This line of reasoning was minimized in the interests of feudal rule, but still existed.
There followed of course the development of Common Law and restrictions on feudal authority until finally the English Parliamentary form was achieved in nearly its modern form. The American Revolution was built on the foundation of that form but took new directions and further extensions of a neglected reasoning.
You have to look at the Declaration of Independence as the model for what later became the national charter, ie: Constitution and Bill of Rights. The Declaration set out the principles of the Revolt; these were several but for our narrow discussion the existence of Rights outside government, which pre-exist and pre-empt governmental law and the absolute right of the People to change their government, by force.
The Bill of Rights was found to be necessary by the framers not due to shortcomings of the Constitution but the inherent shortcomings of government of any sort; that unless certain Rights were noted as guaranteed not granted by government that they would inevitably be infringed. If you look at the BOR in the light of preeminent rights you will see that the Amendment follow the natural course of human behavior, that people will speak their minds, associate, have gods, arm themselves, wish to be secure in their homes, etc. In the Declaration these Rights are attributed to the Creator's design.
When we talk about the 2nd these concepts are its basis, that the right to arms is natural and irrevocable and that the people have recourse to force regarding their government. The wording of the Second becomes very important, and particularly in regard to its meaning in 18th Century America.
"A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed."
"well regulated" in the terminology applied to the military and militia meant "well turned out" and "fully equipped" it did not have the current "ruled" - "regulatory" meaning typical today.
The militia was any able bodied free male (white in cases) etc, in today's world all citizens. The militia specifically was not the organized military, it was the citizenry.
This moves us to a section frequently misused and improperly defined, "a free state," any understanding of the term that uses it to signify a governmental body is deliberately misusing English. These were highly literate and very careful authors, they said exactly what they meant to say, a free state - an indefinite article and no capitalization during a period of "over capitalization." This is of huge significance when the dependent clause "A well...state," is applied to the independent clause "the right..." because the operative noun in the independent clause is the "people." The entire dependent clause is referential to "the people." It is a "free state" of the people, it is a militia of the people, it is a guarantee that the people will have the ability to secure that free state through an un-infringed right to keep and bear arms. It probably is not necessary for this discussion to go through the definitions of arms applicable at the time, other than to note that they would be carried by a single individual, cannons, etc are precluded.
This is the starting point of ANY discussion of the Second, the beginning and its meaning. As with all rights they are not absolute in their application, famously, shouting "Fire!" in a crowded theater. This recognition is one of universal inevitable disastrous outcome with no countervailing value to the individual or people. Fully automatic weapons due to their "spray and pray" characteristics virtually guarantee problems with general availability. Sawed-off shotguns are a fact of law, I believe probably challengeable, but of questionable value to challenge. Their military function is undeniable, their value in self-defense is undeniable, their use in criminal endeavors is minimal. There is little chance of change in that regard and frankly other than as a matter of principle I see little point in pursuing it. Side arms and long guns are of universal military application so we'll stay there.
What about NY,NY? The Fourteenth Amendment clearly imposes the Rights of the Constitution on all jurisdictions. There are no exemptions or exclusions, so the 2nd applies in all cases in all places. (with regard to the "Fire" exceptions-courthouses, etc) NY, NY & DC, et al claim this exception, the problem is that they cannot prove it. They have made the emotional appeal to security and safety without regard to logic or facts. Those laws are blatantly un-Constitutional and unevenly applied - ie: wealth, power, & influence determine the outcome. Concealment of weapons without some sort of permitting process, legitimizing the practice, will have the catastrophic outcomes noted above, law enforcement and the general citizenry should have knowledge of armament in general terms or the security that such carry is approved of by the state. However, such permitting must be evenly applied and restrictive only in regards to lawfulness of usage, not wealth and influence. This applies to "instant background check," the check is that those who have abrogated their rights are not attempting to exercise such. The reason it is IBC is that any other process is an infringement; it is an attempt to unlawfully deny ownership to those lawfully entitled to such.
I'm sure I've left questions unanswered or raised some unaddressed. Huge volumes have been written on this subject, for instance the DOJ's analysis of the 2nd as an individual right vs collective runs over 100 pages with annotations. I've taken a shot at the whole thing in 2pgs.