The Constitution guarantees the right to keep and bear arms individually. The Heller and Mc Donald decisions by the SCOTUS reaffirmed those rights in places like DC and Chicago where there was an outright ban on firearm possession. The Constitution also allows for rights to be limited where there is a need for public safety. Which is why one is required to have a tax stamp for silencers, and fully automatic weapons. Also why felons, people with a history of violence against others and mentally ill are banned from owning firearms.
The limitations in some states and CHL, LTC whatever it is in those states arise from the fact that by carrying a firearm on your person you are more than likely going to be entering into properties which are not yours. For instance in Texas if you are on your property, or in your vehicle, you are not required to have a LTC. The LTC allows you to enter the bank or grocery store or where ever one might go daily and carry your gun. Your rights have limits due to the rights of the property owners rights to their property.
The NICS system which is what is used in conjunction with the 4473 does not require finger prints. But states are required to report court ordered admittance to mental facilities, criminal offenses, ect. This is what is lacking. Many states do not report timely accurate information to the FBI.
The private sales of firearms is not covered by the NICS system. It is the duty of the individual to ensure they are not selling a gun to a prohibited person as best they can. I have only sold one gun individually and that was to a sibling. If I were to sell a gun to someone I wasn’t very familiar with, I would require that person participate in a dealer transfer and fill out the appropriate 4473. That is just me, others may not be so cautious.
As far as rights beginning and ending. If I carry my gun and never infringe on someone else’s right to safety, I haven’t infringed on their right. No different than if I carry words in my mouth and keep it shut from anything that would infringe on another persons rights. If I make a threat to them, or slander them, or do something that would be considered liable, or yell fire in a crowded theater, I have violated my right of free speech and am subject to penalties of law. But we are not required to pass some sort of test in order to exercise our free speech. We are not required to get approval to do so, we are only limited after we do so and violate the law. Many folks think that people should have to pass a test in order to exercise their rights under the 2A, but not any other. That is very curious.