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.