According to the Business Insider article Why 'Star Trek' Has so Much Lens Flare it was all done in-camera and it was all intentional.

"There are so many movies from my childhood that had those that when we were shooting 'Star Trek' I remember saying to Dan Mindel, the DP [director of photography], 'It would be so much fun if we' — I didn't think we were going to have quite that number of them, but it became this thing, and it was ridiculous," he said.

Abrams recalled Mindel bringing in these giant powerful halogen spotlights called "best in show" on set and they would use them while filming.

"It became this weird kind of artform of how to make the perfect lens flares with different kind of lenses," said Abrams. "And it was just this thing that sort of felt like it was a kind of visual system for the movie."

So unfortunately there isn't a version without the lens flare. Abrams did apologize for the overuse of lens flare after his wife complained that the lens flare was so bad in Star Trek Into Darkness she was unable to distinguish who was on screen.

Regarding the difficulty of adding lens flare...

Adding digital lens flare is very simple.

Many common editing software have ways to do it. It is as simple as choosing the point of origin of the lens flare, its type (with respect to the camera recording, like 105mm).

Videocopilot (a nice resource for Adobe After Effects tutorials) even released a plug-in for AAE that allows a user to create many other types of lens flares.

Your questions don't ask about the new Star Trek specifically, but you do mention it in the context for the questions, so I'm going to answer both in general, and for New Trek specifically.

Does the organic picture actually taken by the camera have this lens flare in it? Or is it added to the image in post-production?

Lens flare is caused by the fact that the lenses used in cameras aren't perfect, and so some light is reflected off of the surface of the lens, and thus winds up somewhere the designer of the lens assembly didn't want it to go. Hence the name: it's a flare of light caused by a lens. It can be simulated, though, so there's no guarantee that any particular lens flare you see is natural.

As far as New Trek is concerned, as mentioned in other answers, Abrams liked the look of lens flare, so he had some bright lights brought in to deliberately engender it. It was all done in-camera.

Are there rolls of film/RAW files that don't have it, that would allow a "Director's (well not JJ Abrams) Cut" that dropped it?

Most people don't like lens flare. So, in the general case, probably 99% of lens flare you see is "natural", unavoidable flare that's recorded directly to the raw film.

In the case of New Trek, as mentioned above and in other answers, all the flare was done with actual lights on-set, so it can't be removed.

How hard would it be to digitally add lens flare to shows that don't have it, like Downton Abbey or The Orville?

All shows that were recorded with a real camera (i.e. not computer-generated or otherwise animated) have lens flare. It's just a side effect of living in an imperfect world. It's just not noticeable in most shows, because the lighting guys take steps to avoid and minimize it. But its incredibly easy to add in post, if you decide your scene doesn't have enough. Most video editing software can add it automatically. It's also simple enough to do manually.

Old cinematographer here. Above answers are partially correct. History of lens flare:

1. First lens flares were undoubtedly accidental (light reflecting between elements of lenses).
2. Somebody liked the fx, so carefully set up camera angle so lens would capture light in this way.
3. Flare filters were produced, that attached to camera lens, so vitually any shot could incorporate a flare.
4. Now, software can add a flare anywhere, in any scene, even in total darkness, and after the scene is shot.

As with most sfx nowadays, it takes little skill, which is why filmakers like Scorcese scorn movies that rely on technical stuff to impress.