There is a problem with your question which is at the heart of many misunderstandings about law in the US.
Citizens can do everything and more that the police can do, because the government is beholden and made up of the Citizens.
If you watch a police officer and they do x, you can do x too. The difference is that they have licenses and government issued warrants and such that allow them to do a few more things legally than you. It's not because you couldn't go through the same processes and get those approvals, but because part of those checks are an insurance to everyone that procedures have been carried out to make sure these things are on the up and up and they are held accountable.
Let's take Superman or Spiderman. They are not breaking any laws when there is a bank robbery and they intervene. They are well within their legal rights to do that and being that the cops would see how useful and helpful this is they'd work with them. They are government agents, because that's how the US is, but you mean they are not official representatives. I don't see that as an issue. There is no reason that they have to be official reps and this goes into the 3rd point...
Everyone can arrest someone else, etc. You, as a member of the US citizenry have the ability to arrest people, ie. detain people when they have commited a crime. Just like the police. The difference is in how much leeway we give the police on this matter, because it is illegal to detain someone forcibly. Both you and the police can be sued and criminally charged if it is shown not to be justified, but the police can get away with it longer, simply because they have a more reasonable argument that it is justified, given it's their job.
What IS vigilantism is more akin what Batman does (though you can debate it, but we're just going for simplicity sake). Batman looks for clues, enters into peoples' property, and actively starts situations with people who are possibly criminals.
I don't know how far Detectives can push to enter property and such because I haven't looked into it, but let's say you can get a license to do that. The primary difference between what you as a citizen can do and what the cops can do is start a situation. The police can go to a wherehouse that has criminals in it, bust in and arrest everyone. You cannot. However, if police bust in a place and start arresting people you can help. Likewise if you hear or see something going on you can interfere.
The primary reason that "Heroes" are not welcome in the real world is that they don't have infrastructure, cause more problems than help due to this, and are generally less easily held accountable through the insurrances we have created. If real Superheroes started appearing you have to extend some of the assumptions of a police officer doing their job to the Superhero, such as detaining people for longer.
The main issues are really augmentations that you might have to make to certain things, such as can Superman ever really be a Vigilante? Remember, what makes a vigilante is initiating the situation without seeing or hearing a justifiable reason to interfere. Superman can hear everything. Superman can naturally see through walls. Is it illegal for Superman to bust into a warehouse where victims are being held? What if there is a gun pointed at them? A cop wouldn't know. A normal person wouldn't know so to them the answer is that they can be vigilantes, but if they did it it would be illegal, but if Superman did it it wouldn't under our laws, because he saw it.
What about people like Cyborg? They don't just have internet access, they have a direct access to the signals that are floating in the air and he naturally decrypts them without any special thing going on. You need warrants and such to do this type of stuff, because you wouldn't "normally" be able to see/access that stuff and it is presumed "private", but Cyborg does it without thinking. Is it then not illegal for him to have such access?
What about someone that could time travel, but not interfere except in the present or could read minds? Where does their justifiable interference stop? Where does them "seeing" things stop. If I can see you're entire history from the moment your born till present and know the crimes you have commited and all those type things, as our laws are I would be 100% justified interfering at any point in your life that I so desire because you're the one initiating the action from my perspective and I have reasonable cause to stop and arrest you.
On the other side of the spectrum you have to establish whole new rules of the concept of escalation of force models. Superman as a hero is under no danger from the average citizen, but just about any action he takes is clear a level of force that would probably lie outside easily definable boudries of force. On the other hand, an evil superman, when is deadly force reasonable? Superman can kill you with a punch from the other side of the Earth before you can unholster your gun and discharge 1 round, and it wouldn't even be lethal to him. Does that mean then that any interaction with Superman that there nothing that you can do (with normal weapons/actions) that would ever be considered illegal?
Those type of boundries would probably be established along with the extension of presumptive abilities of police onto Supers. But other than these modifications to current law every point you made is perfectly within the legal bounds of civilians today. It's just discouraged due to various accountability and safety issues.
Also, as far as the idea of Superheroes in general not working with police/military. I find that harder to believe in most cases if we're talking abilities in line with x-men and lower tier heroes, because working with the police as part of the police would just be, overall, better due to infrastructure, support, public scrutiny, economics, etc. Superman doesn't need that and working "with" the police would slow him down. Batman has that through an elite selection and training program + money. Nightcrawler, Rogue, Jubilee, Mystique, Cyclops, Spiderman, Captain America, Iron Man, etc do not. The issue there is then 16 year old Peter Parker, bit by radioactive spider and gets all these powers that can be very useful, but he's 16, so he can't join the Police/Military, but he certainly won't be sitting around doing nothing with his powers, so what do you do with him?
After a while what you really will end up getting is that you'll get training facilities and powers as being a natural part of your medical records, and perhaps licenses for using powers of various types in public and you might get a society that is quite a bit different in certain aspects, for example, clothing. People might go the route of "you can't use your see through anything vision under any circumstance unless you're doing x" or they might realize they can't control that and to try to would be anti-freedom and as a result wear less clothes, because what's the point if random people can see through it anyways.
Hope that all helps to answer your question. It's not really a special development that would happen to make a society act in the way that you're talking about, because our society already acts in that way. Its just not apparent cuz we don't have Supers running around (for the most part... I mean have Zombie-esque type people in Florida and that's pretty Super)