In this world I'm working on, magic is extremely bound by the laws of physics. Magic is explicable through science, and many people study it. The study of magic itself is like studying physics; the study of applied magic is like engineering. I like to think of magic as behaving similar to electromagnetism. Humans emit electromagnetic radiation naturally, but it would be very unnatural if a human suddenly could shoot out a bolt of lightning, or if they could use their bare hands to power a circuit.
In a world like that, where magic is akin to electromagnetism, how would a civilized society react to individuals capable of magical feats? Note that these mages would be incredibly rare--perhaps 1 for every 750,000.
Society would view this as a threat and eliminate it entirely.
Society, already advanced in studying magic, wishes to learn more and wants to determine if the condition can be induced onto others.
Society finds a means of using their abilities to conduct war and trains them as soldiers.
I can see a lot of possibilities. I want to know: which possibility, of the ones above or otherwise, is the most realistic one to occur, and why?
1: Does magic run in families or is it totally random?
Let's say that it is beyond the scope of genetics. It is not inherited. No single race or ethnicity is more likely than another to have mages appear. Mages may also appear in animals, but the scope of ability is limited based on physical constraints. A cow's magic is different than a human's.
2: Is it a recent thing? Did mages start appearing like X-Men or have they always been?
It is not a recent thing, but it gets more noticeable as time goes on due to ease of communication amongst people. Additionally, since the probably of being born a mage remains the same throughout time, having an increased population means the likelihood of mages appearing also increases over time.
3: Do mages have any historical organization/guild/power base?
I doubt this largely because of how rare it is, though I can see these emerging as time goes on, for reasons stated in 2.
4: Just how powerful are they? World shattering, city shattering or small country village shattering?
This I believe varies greatly depending on the individual. While some mages may only be able to, say, cause light to emerge from their hands, others may very well be capable of devastating offensive tactics. That said, since I do want magic to be at least partially tied down to a scientific study, I wouldn't imagine an individual capable of destroying the world. For cities, it'd likely have to be a small one. Humans have limits, and as such human mages also have limits.
All three. Here in the US you would likely get all three in one country. The average person would probably fear them, some would find them fascinating, elevating them to the level of super-stars, 'scientists' would want to study them, and governments would want to control them and use them as a threat or worse.
Re the answers: since they've existed historically I'd say that puts them in a different light. Mages aren't going to be in competition with governments: historically they've been governments or a major part of the power base of governments.
But no dynasties since it doesn't run in families. If it did then mages would be the dominant group with almost all royal families being very magical.
In fact I can imagine that history would look very different. Before the invention of guns/sniper rifles mages would have been the dominant force in any war. Since it's totally random and 99% of the population would have been dirt poor every generation almost all of the most powerful people in the world would would have risen from poverty to the heights of power. Any nobility that abused the poor would be facing a new ruler who had been born in poverty within a few decades.
Social mobility is going to look quite different and there's unlikely to have been a history of long-term royal lines. Indeed such a world is likely to have been far more egalitarian out of practical and justified fear, any big country is going to produce a few mages and most will be born the children of peasants.
For stability it's likely that mages will seek out others like themselves to form some kind of guild in order to get some kind of continuity of power and to make sure their grandkids aren't slaughtered by the next generation of mages.
2 and 3 are certain though whether mages are the ones controlling it is uncertain.
Finding mages at a young age when they can still be indoctrinated will have become part of the culture but they'll have been scooped out of the regular population so predictably that social views towards them would probably be highly negative.
A countries mages are likely to be viewed as a major military asset but if they've been part of the system for many centuries they're likely to have status as well.
Also there will be a lot of chicken mages. Worldwide over 50 billion chickens bred every year so that well over 50,000 chicken-mages. If their power can be tapped it will be.
A civilised society would probably become a mix of scared, angered, confused, curious, interested and intrigued. Here are the reasons for each of these: Scared/angered because this magic could easily be misused (depending on how destructive this feat of magic was). Confused/curious/interested because of what just happened (depending on what exactly happened though). Intrigued because of the abnormality of what just happened (depending on how dramatic this magic was).
Another thing that would probably happen is if the police equivalent in your society (assuming there is a police equivalent) are there during the feat there will probably be actions taken to restrict the person who did the magic (again depending on how dramatic and how destructive this magic was) to ensure safety of the public. What exactly would happen to the person after that would mostly depend on what sort of police exists in your society (are the police more aggressive and brutal or are they more calm), how dramatic and destructive this magical feat was and what the witnesses say as well as the "magic guy"'s attitude/reaction towards the police.
If the police weren't there at the time but were notified about it by a decent amount of people then an investigation would probably take place. If the police are ultra brutal and aggressive then they might take a much more direct approach like searching for this person house by house.
I can't be too specific because a lot of this depends on the type of society it is and specifics but hopefully this narrows it down a bit.
EDIT: I replied before you editted the question to include the three different thoughts.
First of all, 1:750k isn't super-rare. Europe has almost 750 million people currently. So there would be around a thousand of your mages in Europe alone, and ten times as many globally.
That is more than enough to form their own society, organisations, maybe even small country.
The answers to your question depend a lot on whether or not this happens. You evade the question with your answer 2, but it is absolutely vital to the entire concept.
If they don't form a guild, the governments of the world have more than enough means to control them. Some would use positive reinforcement (honours, loyalty, etc.) some would use soft pressure (prejudices in society, cultural taboos), some hard pressure (laws, regulations, punishments) and some suppression (jail, execution).
Society would react much like it does to anything else. Drugs, climate change, disasters, new technology - with a variety of responses.
If the mages cooperate and form an organisation, they can free themselves from outside influence. They would become a player in global politics instead of pawns - with all the opportunities and risks associated.
If they have a physical location, they need to be ready for war. Someone will want to fight them, for religious or other reasons. If they don't have a physical location, they will be in the place of other international organisations, and can potentially be restrained by laws or sanctions.
Many possibilities are discussed in the TV Tropes page for "Super Registration Act": http://tvtropes.org/pmwiki/pmwiki.php/Main/SuperRegistrationAct .
One particular one which seems likely to me is that mages would not be explicitly forbidden to use their magic, but would be so tightly regulated that magic use became next to impossible in practice. You'd have to submit an application in triplicate three months in advance before you cured a wart.
A good effect of this from the government's point of view would be that mages would be willing to join the government research programme or the army simply to get a chance to use their powers - without the government having to introduce conscription, or fend off civil rights lawyers.
How would magic affect the subjects taught in school?