Do not point your gun at anyone or anything if you are not willing to accept the responsibility, and legal consequences, if you pull the trigger. This is the very first rule of gun safety. I will repeat this often in my answer because it bears repeating often.
So, for existing law:
DC v. Heller determined that the Second Amendment allows for the defense of oneself on one's own property. As a general rule, this holds IFF the only conditions are met: 1. The person is an intruder in your home. 2. You cannot safely flee your home.
In most states, condition 2 is dropped (Only Vermont and D.C. which have no law, require condition 2 in theory... case law will determine the outcome.). You need not prove that the intruder is about to commit a felony on you, only that you were within reason to attack that person. Per your questioned states: North Carolina has an explicit carve out for Police and Bail Bondsmen acting with lawful authority and who are complying with all laws.
With all Castle Laws, if you invite the person you pull the gun on into your house then they are not their aggressively (at least, until they are asked or intend to leave. Certainly a 3 am entry by your neighbor is unlawful if he left at 10pm).
North Carolina and Tennessee are also "Stand-Your-Ground" States, which allow for deadly self-defense of yourself or another if you are not an aggressor and are not acting in an unlawful manor and not engaged in trespassing. In other states, they usually require a duty to flee in public settings before you can use lethal force for self-defense (i.e. If you can get away safely from the criminal, you should do so first. Otherwise, you can use lethal force.).
In any case where the person you pull a gun on is not an aggressor and not reasonably indicating they are committing a crime, pulling a gun on someone is considered assault with deadly weapon. Even if you do not fire... even if it's a toy realistic gun with the orange safety cap taken off or not visible... even if it looks like a gun and the guy cannot tell.
In your initial case, it would be an illegal use of your gun because you have not provided me with any evidence that the person was in your house because of trespass or that they were reasonably committing an illegal act (This is technically only. I need you to show me that this person is on your property without your authorization).
In your first variation, again, if you can prove that the above scenario was legal, threaten to shoot him is fine. You are defending yourself against him and his actions. You need not kill a criminal to defend yourself. It may actually help your case if he still decides to act even with this warning.
In the second variation, again it depends. A few states extend Castle Doctrine to your car, so if you are broken down on a public the highway and need to defend yourself from unlawful entry to your vehicle, the same rules apply if you are inside your car. If you are on your private property and the morning jogger is your target, you need to make sure he is reasonably threatening you or another person. Depending on the nature this will apply to Castle Doctrine or Stand-Your-Ground, though if he's coming at you while you are on your property, the former may apply. If he is going for the neighbor I would err to the latter. Needs more detail.
In your final situation, NC and TN are both Stand-Your-Ground so presuming you are not the aggressor those rules apply.
If the gun holder can not reasonably prove his claim to self-defense, pointing a gun at a person is Assault even if the gun could never fire a bullet (it's not loaded, it's broken, it's a realistic toy and the orange tip on the barrel was removed or not visible to the guy you point it at.). If you fired it could be Murder if the guy dies.